{"paper_id":"0822fb77-99b8-4177-b523-bdb97465bba5","body_text":"NMDA Receptor Kinetics Drive Distinct Routes to1\nChaotic Firing in Pyramidal Neurons2\nMehdi Borjkhani 1,2∗, Hadi Borjkhani 3,4, Morteza A. Sharif 5,\nFariba Bahrami6, Mahyar Janahmadi 7\n1International Centre for Translational Eye Research (ICTER), Institute of Physical Chemistry,\nPolish Academy of Sciences, Warsaw, Poland\n2Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland\n3Faculty 1, University of Applied Sciences (HTW) Berlin, Berlin, Germany\n4Intelligent Biomedical Sensing (IBS) Lab, Machine Learning Department, TU Berlin, Berlin, Germany\n5Optics and Laser Engineering Group, Faculty of Electrical Engineering,\nUrmia University of Technology, Urmia, Iran\n6CIPCE, Motor Control and Computational Neuroscience Laboratory,\nSchool of ECE, College of Engineering, University of Tehran, Tehran, Iran\n7Neuroscience Research Center and Department of Physiology, Medical School,\nShahid Beheshti University of Medical Sciences, Tehran, Iran\n∗Corresponding author: mborjkhani@ichf.edu.pl\n3\nAbstract4\nNeuronal firing patterns emerge from complex interactions between intrinsic membrane prop-5\nerties and synaptic receptor dynamics. N-methyl-D-aspartate (NMDA) receptors critically shape6\ncalcium influx and synaptic plasticity through their voltage-dependent Mg2+ block and prolonged7\nactivation kinetics. We developed a Hodgkin-Huxley-type computational model incorporating8\nNMDA, AMPA, and GABA receptor kinetics to investigate how NMDA receptor closing rates9\n(βN M DA) and glutamatergic stimulation frequency control neuronal dynamics.10\nSystematic analysis of 2,942,093 inter-spike intervals across 1,961 parameter combinations11\nrevealed two mechanistically distinct pathways to firing irregularity. Pathway 1 involves rapid12\nNMDA deactivation (βN M DA > 0.06 ms−1) at elevated stimulation frequencies, producing deter-13\nministic chaos with compromised information encoding (entropy: 1.441 bits, mutual information:14\n0.185 bits). Pathway 2 results from slow NMDA deactivation (βN M DA < 0.02 ms−1) under weak15\ndrive, creating irregularity through prolonged receptor activation and sustained calcium influx16\n(entropy: 1.347 bits). An optimal kinetic window emerged at βN M DA = 0.028 ms−1, maximizing17\ninformation transfer (0.275 bits) while maintaining stable dynamics.18\nEntropy-Lyapunov correlation analysis confirmed deterministic chaos (r = 0.150, p ¡ 0.001).19\nFrequency-dependent chaos onset thresholds demonstrated systematic erosion from 0.000 ms −120\nat low frequencies to 0.150 ms −1 at high frequencies. GABAergic inhibition provided frequency-21\nselective stabilization, expanding stable parameter space by 34.222\nThese findings establish NMDA receptor kinetics as fundamental controllers of cortical ex-23\ncitability and information processing. The dual-pathway framework provides mechanistic in-24\nsights into addiction-related memory formation, where prolonged NMDA activation enables25\npathological plasticity, and visual processing disorders, where altered kinetics disrupt retinal26\nfunction and cortical oscillatory balance. The identification of optimal kinetic windows and27\n1\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nfrequency-selective GABA modulation suggests therapeutic strategies targeting kinetically-specific28\ninterventions for neuropsychiatric disorders involving NMDA dysfunction.29\nKeywords: NMDA receptors, neuronal dynamics, synaptic plasticity, chaos theory, infor-30\nmation theory, computational neuroscience, addiction, visual processing31\n1 Introduction32\nInformation processing in the brain depends on precisely tuned interactions between intrinsic mem-33\nbrane conductances and synaptic receptor dynamics. NMDA receptors play a pivotal role in exci-34\ntatory synaptic transmission, distinguished by their voltage-dependent Mg 2+ block and relatively35\nslow deactivation kinetics Kampa et al. (2004); Vargas-Caballero and Robinson (2004). These36\ncharacteristics mediate extended calcium influx into postsynaptic neurons, influencing long-term37\npotentiation (LTP), long-term depression (LTD), and other plasticity-related processes L¨ uscher38\nand Malenka (2012). Beyond synaptic plasticity, NMDA receptors modulate neuronal firing stabil-39\nity and variability, with direct consequences for information encoding and transmission Harsch and40\nRobinson (2000); Hunt and Castillo (2012).41\nA central question in computational neuroscience concerns how changes in NMDA receptor gat-42\ning kinetics—particularly closing rates (βN M DA) and glutamate stimulation frequencies—transition43\nneurons between stable periodic firing and irregular chaotic regimes Durstewitz and Gabriel (2007);44\nSoudry and Meir (2012). While irregular spiking can degrade information transfer reliability, it may45\nalso increase dynamic range and encoding flexibility de Ruyter van Steveninck et al. (1997); Ermen-46\ntrout et al. (2008). Despite extensive research on excitatory-inhibitory balance effects on neuronal47\ndynamics, the detailed parameter space of NMDA receptor kinetics has received limited systematic48\ninvestigation. The precise influence of NMDA receptor kinetics on plasticity induction, inter-spike49\ninterval (ISI) frequency band emergence, and their implications for single-cell information encoding50\nremains incompletely understood.51\nThe clinical significance of NMDA receptor kinetic alterations extends across multiple neuropsy-52\nchiatric conditions. In schizophrenia, altered receptor kinetics may contribute to gamma oscillation53\nabnormalities and cognitive deficits Coyle (2012); Gandal et al. (2012). Autism spectrum disorders54\ninvolve NMDA-mediated excitation-inhibition imbalances affecting sensory processing and social55\ncognition Rubenstein and Merzenich (2003); Lee et al. (2017). Alzheimer’s disease features pro-56\ngressive NMDA receptor dysfunction correlating with synaptic loss and memory impairment Wang57\nand Reddy (2017); Snyder et al. (2005); Hynd et al. (2004). Chronic pain conditions exhibit altered58\nNMDA kinetics in spinal cord circuits, contributing to central sensitization Latremoliere and Woolf59\n(2009); Zhuo (2016).60\nIn addiction neuroscience, chronic substance exposure alters NMDA receptor expression and61\nkinetics, contributing to pathological synaptic plasticity underlying addiction-related memory for-62\nmation and relapse vulnerability Kalivas (2009); Wolf (2016). Pathological memory formation63\nrepresents a critical mechanism underlying addiction persistence and relapse vulnerability. Unlike64\nnormal learning and memory, drug-induced memories exhibit abnormal persistence, resistance to65\nextinction, and heightened salience that can trigger craving and relapse even after prolonged absti-66\nnence Kalivas (2009); Wolf (2016). These pathological memories form through aberrant synaptic67\nplasticity mechanisms involving dysregulated NMDA receptor function and downstream signaling68\ncascades, particularly calcium-dependent processes such as CaMKII phosphorylation L¨ uscher and69\nMalenka (2011); Pascoli et al. (2014).70\nPrevious computational modeling has demonstrated that opioid exposure fundamentally alters71\nsynaptic plasticity mechanisms through disruption of calcium homeostasis and CaMKII phosphory-72\nlation dynamics Borjkhani et al. (2018a,b). These studies revealed that chronic drug exposure shifts73\n2\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nthe balance between LTP and LTD, favoring persistent synaptic modifications that encode drug-74\nassociated memories. NMDA receptors serve as critical gatekeepers in this process, with their kinetic75\nproperties determining calcium influx magnitude and persistence. Our research group has previ-76\nously developed computational frameworks demonstrating how opioids induce pathological memory77\nformation through theta rhythm generation during chronic consumption (Borjkhani et al., 2018c),78\nand how cocaine disrupts action potential generation by reducing potassium currents (Borjkhani79\net al., 2022). These studies established the foundation for investigating drug-induced alterations in80\nneural dynamics and synaptic plasticity mechanisms.81\nEvidence suggests NMDA receptor antagonists can disrupt drug memory reconsolidation and82\nreduce relapse rates in preclinical studies Lee et al. (2006); Das et al. (2013). However, the precise83\nrelationship between NMDA receptor kinetic properties and pathological memory formation versus84\nmaintenance remains incompletely characterized. This knowledge gap represents a barrier to devel-85\noping targeted therapeutic interventions that could selectively disrupt maladaptive memories while86\npreserving normal memory function.87\nIn visual neuroscience, NMDA receptors are essential for retinal ganglion cell function and88\ncortical visual processing Manookin et al. (2010); Fox and Daw (1992). Receptor dysfunction con-89\ntributes to glaucoma, diabetic retinopathy, and cortical visual impairment through altered firing90\npatterns that disrupt normal visual processing Bai et al. (2013); Seki and Lipton (2008). In pri-91\nmary visual cortex, NMDA receptors mediate orientation selectivity refinement, ocular dominance92\nplasticity, and contrast adaptation mechanisms Morishita and Hensch (2008); Hensch (2005). The93\nkinetic properties of these receptors directly influence critical period timing and the capacity for94\nexperience-dependent plasticity throughout life Hensch (2005).95\nRecent evidence suggests altered NMDA kinetics contribute to visual processing deficits in neu-96\nrodevelopmental disorders. In amblyopia, disrupted NMDA-dependent plasticity prevents normal97\nbinocular integration, while in autism spectrum disorders, altered excitation-inhibition balance af-98\nfects visual motion processing and gamma oscillations Foss-Feig et al.(2013); Robertson and Baron-99\nCohen (2016). Age-related changes in NMDA receptor function may contribute to declining visual100\nprocessing efficiency in older adults, with reduced NMDA-mediated plasticity affecting contrast101\nsensitivity and temporal processing Hua et al. (2008).102\nHere, we develop and analyze a single-compartment Hodgkin-Huxley-type model of a pyramidal103\nneuron incorporating sodium, potassium, and calcium currents, along with GABAergic, AMPA, and104\nNMDA synaptic conductances. Building upon our previous computational investigations of opioid-105\ninduced memory formation (Borjkhani et al., 2018c) and cocaine effects on neuronal excitability106\n(Borjkhani et al., 2022), we extend this framework to systematically examine NMDA receptor107\nkinetic control of cortical dynamics. We include CaMKII phosphorylation dynamics as a biochemical108\npathway linking calcium transients to synaptic plasticity. By systematically varying βN M DA and109\nglutamatergic drive frequency, we evaluate how neuronal firing evolves through different oscillatory110\nmodes and assess impacts on spike-timing entropy, maximum Lyapunov exponent, and mutual111\ninformation. We investigate how GABAergic inhibition modulates these transitions through fast112\ninhibitory currents that control neuronal excitability and chaotic dynamics.113\nOur findings reveal dual pathways to firing irregularity and establish NMDA receptor kinetics114\nas fundamental controllers of cortical excitability and information processing. These results provide115\nmechanistic insights into addiction-related memory disorders and visual processing dysfunction,116\nsuggesting therapeutic strategies targeting NMDA receptor kinetic normalization and oscillatory117\npattern restoration.118\n3\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n2 Materials and Methods119\nFigure 1 provides an overview of our comprehensive computational approach to investigating NMDA120\nreceptor kinetics and their control over neuronal dynamics. The workflow encompasses four in-121\ntegrated stages: systematic model development and parameter space exploration, application of122\nmultiple dynamical analysis methods, identification of key mechanistic findings through the dual123\npathways framework, and translation to clinical applications. This systematic approach allows us124\nto bridge molecular receptor properties with network-level dynamics and establish direct relevance125\nto addiction neuroscience and visual processing disorders. The following sections detail each com-126\nponent of this workflow, beginning with the neuronal model architecture.127\n2.1 Neuronal Model Overview128\nFor the simulation of a pyramidal neuron in the cortex, we employed a single-compartment Hodgkin-129\nHuxley-type model based on Golomb et al. Golomb et al. (2006). The model was modified to130\nincorporate glutamatergic and GABAergic receptors (NMDA, AMPA, and GABA receptors) along-131\nside ionotropic channels, enabling responses to both excitatory and inhibitory neurotransmitters.132\nFurthermore, the model incorporates CaMKII phosphorylation dynamics as a function of calcium133\nconcentration variations, providing a mechanistic link between synaptic activity and plasticity Bor-134\njkhani et al. (2018a). This computational framework allows investigation of NMDA receptor kinetics135\neffects on neuronal excitability relevant to addiction-related plasticity mechanisms and visual pro-136\ncessing disorders. Figure 2 shows the general structure of the modeled elements.137\n2.1.1 Model Validation and Verification138\nThe model was validated against experimental data from pyramidal neurons in layers 2/3 of visual139\ncortex Markram et al. (1997). Key validation metrics included: (1) resting potential (-65 ± 5 mV),140\n(2) action potential amplitude (80-100 mV), (3) spike threshold (-45 ± 3 mV), and (4) adaptation141\nratio (0.6-0.8) during sustained current injection. Numerical integration was verified using analytical142\nsolutions for simplified cases and compared with NEURON simulator results (relative error< 0.1%).143\n2.2 Membrane Dynamics144\nThe membrane potential ( V ) in the excitatory postsynaptic neuron is governed by the following145\ndifferential equation:146\nCm\n∂V\n∂t = − (INa + INaP + IKdr + IA + IM + ICa + IC + IsAHP + IL + Isyn) (1)\nwhere the membrane capacitance Cm = 1 µF/cm2 and the total synaptic current is:147\nIsyn = IAMPA + INMDA + IGABA (2)\n= gAMPAmAMPA(V − EAMPA)\n+ gNMDAmNMDABMg(V )(V − ENMDA)\n+ gGABAmGABA(V − EGABA) (3)\n4\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n2.3 Intrinsic Ionic Currents148\n2.3.1 Sodium Currents149\nThe transient sodium current is described by:150\nINa = gNam3h(V − ENa) (4)\nwhere gNa = 35 mS/cm2 and ENa = 55 mV is the sodium reversal potential. The activation and151\ninactivation gating variables are denoted by m = m∞(V ) and:152\ndh\ndt = ϕ[h∞(V ) − h]\nτh(V ) (5)\nwhere ϕ = 5 is a temperature factor and the time constant is:153\nτh(V ) = 0.1 + 0.75 · [1 + exp(−(V + 40.5)/(−6))]−1 ms (6)\nThe persistent sodium current is:154\nINaP(V ) = 0.1 · p∞(V ) · (V − 55) mS/cm2 (7)\n2.3.2 Potassium Currents155\nThe delayed rectifier potassium current is:156\nIKdr(V, n) = gKdrn4(V + 90) (8)\nwhere gKdr = 6 mS/cm2 is the default conductance. The gating variable n follows:157\ndn\ndt = ϕ[n∞(V ) − n]\nτn(V ) (9)\nwith time constant:158\nτn(V ) = 0.1 + 0.5 · [1 + exp(−(V + 27)/(−15))]−1 ms (10)\nThe A-type potassium current has the following dynamics:159\nIA(V, b) = 1.4 · a2\n∞(V ) · b · (V + 90) mS/cm2 (11)\nwhere a = a∞(V ) and:160\ndb\ndt = b∞(V ) − b\n15 ms (12)\nThe muscarinic-sensitive potassium current is:161\nIM(V, z) = 1 · z · (V + 90) mS/cm2 (13)\nwith gating variable dynamics:162\ndz\ndt = z∞(V ) − z\n75 ms (14)\n5\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n2.3.3 Calcium Currents163\nThe high-voltage calcium current is:164\nICa(V, r) = 0.2 · r2 · (V − 120) mS/cm2 (15)\nwith gating variable:165\ndr\ndt = r∞(V ) − r\n1 ms (16)\nThe fast calcium-activated potassium current is:166\nIC(V, c) = 10 · d∞([Ca2+]i) · c · (V + 90) mS/cm2 (17)\nwhere [Ca2+]i is the intracellular calcium concentration, and:167\ndc\ndt = c∞(V ) − c\n3 ms (18)\nd∞([Ca2+]i) = [1 + 6/[Ca2+]i]−1 (19)\nThe slow calcium-activated potassium current (afterhyperpolarization) is:168\nIsAHP(V, q) = 5 · q · (V + 90) mS/cm2 (20)\nwith gating variable dynamics:169\ndq\ndt = q∞([Ca2+]i) − q\n450 ms (21)\nq∞([Ca2+]i) = [1 + 24/[Ca2+]4\ni ]−1 (22)\n2.3.4 Leak Current170\nThe leak current is modeled as:171\nIL = 0.05 · (V + 70) mS/cm2 (23)\n2.4 Calcium Dynamics172\nThe intracellular calcium concentration evolves according to:173\nd[Ca2+]i\ndt = −0.13ICa − 0.012INMDA − 0.0012IAMPA − [Ca2+]i\n13 ms (24)\nThis equation accounts for calcium influx through voltage-gated calcium channels, NMDA receptors,174\nand AMPA receptors (to a lesser extent), as well as calcium extrusion and buffering mechanisms175\nwith a time constant of 13 ms.176\n2.5 Synaptic Currents177\n2.5.1 AMPA Receptors178\nThe AMPA-mediated current is calculated as:179\nIAMPA = gAMPA · mAMPA · (V − EAMPA) (25)\nwhere gAMPA = 0.5 nS and EAMPA = 0 mV. The gating variable mAMPA follows:180\ndmAMPA\ndt = αAMPA · Gglu(t) · (1 − mAMPA) − βAMPA · mAMPA (26)\nwhere αAMPA = 1.1 mM−1ms−1 and βAMPA = 0.67 ms−1 are the opening and closing rates, respec-181\ntively. Gglu(t) represents the time-varying glutamate concentration that activates the receptor.182\n6\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n2.5.2 NMDA Receptors183\nThe NMDA current is modeled as:184\nINMDA = gNMDA · mNMDA · BMg(V ) · (V − ENMDA) (27)\nwhere gNMDA = 0.5 nS and ENMDA = 0 mV. The voltage-dependent Mg 2+ block is:185\nBMg(V ) = 1\n1 + [Mg2+]o\n3.75 exp(−0.062V )\n(28)\nwhere [Mg2+]o = 2 mM is the extracellular magnesium concentration.186\nThe NMDA receptor gating variable follows:187\ndmNMDA\ndt = αNMDA · Gglu(t) · (1 − mNMDA) − βNMDA · mNMDA (29)\nwhere αNMDA = 0.14 mM−1ms−1 is the opening rate and βNMDA is the systematically varied closing188\nrate (range: 0.01–0.1 ms −1). This parameter represents the key experimental variable in our study,189\nas alterations in NMDA receptor kinetics are implicated in both addiction-related plasticity and190\nvisual processing disorders.191\n2.5.3 GABA Receptors192\nThe GABAergic inhibitory current is:193\nIGABA = gGABA · mGABA · (V − EGABA) (30)\nwhere gGABA = 1.0 nS and EGABA = −70 mV.194\nThe GABA receptor gating variable follows:195\ndmGABA\ndt = αGABA · GGABA(t) · (1 − mGABA) − βGABA · mGABA (31)\nwhere αGABA = 2.0 mM−1ms−1 and βGABA = 1.5 ms−1.196\n2.6 Activation Functions197\nAll steady-state activation and inactivation functions follow the standard Boltzmann form:198\nx∞(V ) = [1 + exp(−(V − θx)/σx)]−1 (32)\nwhere x can be replaced by m, h, n, a, b, z, p, r, or c. The parameters are summarized in Table 1.199\n2.7 CaMKII Phosphorylation Dynamics200\nPostsynaptic Ca2+ concentration variations lead to CaMKII phosphorylation, which is governed by201\nequations adapted from Borjkhani et al. Borjkhani et al. (2018a,b) and Zhabotinsky Zhabotinsky202\n(2000):203\nPh.CaMKII = fCaMKII([Ca2+]i) (33)\nThe detailed phosphorylation cascade involves 10 differential equations governing the concen-204\ntrations of i-fold phosphorylated CaMKII (P i, where i = 0, 1, ...,10). The system includes:205\n7\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nTable 1: Gating Variable Parameters for Activation/Inactivation Functions\nVariable θ (mV) σ (mV) Source\nm -30 9.5 Golomb et al. (2006)\nh -45 -7 Golomb et al. (2006)\nn -35 10 Golomb et al. (2006)\na -50 20 Golomb et al. (2006)\nb -80 -6 Golomb et al. (2006)\nz -39 5 Golomb et al. (2006)\np -41 3 Golomb et al. (2006)\nr -20 10 Golomb et al. (2006)\nc -30 7 Golomb et al. (2006)\nPhosphorylation rates:206\nv1 = 10k1([Ca2+]/KH1)8P0\n(1 + ([Ca2+]/KH1)4)2 (34)\nv2 = k1([Ca2+]/KH1)4\n1 + ([Ca2+]/KH1)4 (35)\nv3 = k2ep\nKM + P10\ni=1 iPi\n(36)\nwhere k1 = 0.5 s−1 is the I1-dependent regulation rate of PP1, KH1 = 4 µM is the Hill constant207\nof CaMKII for calcium activation, KM = 20 µM and k2 = 10 s−1 are the Michaelis and catalytic208\nconstants, respectively.209\nAdditional parameters:210\n• ep: PP1 concentration not bound to I1P (active protein phosphatase)211\n• ep0 = 0.1 µM: total PP1 concentration212\n• I0 = 0.1 µM: free I1 concentration213\n• k3 = 1 µM−1s−1 and k4 = 10 −3 s−1: association and dissociation rate constants of PP1-I1P214\ncomplex215\n• vCaN = 2 s−1: rate of I1P dephosphorylation due to calcineurin216\n• vPKA = 0.45 µM/s: phosphorylation rate of I1 due to PKA217\n• KH2 = 0.7 µM: calcium activation Hill constant of calcineurin218\nThe total phosphorylated CaMKII is:219\nPh.CaMKII = fCaMKII([Ca2+]i) =\n10X\ni=1\nPi (37)\nCaMKII phosphorylation levels were recorded at each time step and correlated with ISI patterns220\nto investigate plasticity-related dynamics relevant to addiction and visual processing mechanisms.221\n8\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n2.8 Computational Implementation222\n2.8.1 Software and Hardware Environment223\nAll simulations were implemented in Python 3.8.10 using NumPy 1.21.0, SciPy 1.7.0, and Mat-224\nplotlib 3.4.2. Computations were performed on a high-performance computing cluster with Intel225\nXeon Gold 6248 processors (2.5 GHz, 40 cores) and 192 GB RAM. Parallel processing utilized the226\nmultiprocessing library with 20 worker processes. Total computational time was approximately 480227\nCPU-hours.228\n2.8.2 Numerical Integration and Simulation Parameters229\nAll simulations employed fourth-order Runge-Kutta integration with ∆ t = 0.05 ms, chosen based230\non convergence analysis showing < 0.1% error compared to ∆t = 0.01 ms. Simulation duration was231\n10 seconds with the first 2 seconds discarded to eliminate transients, determined from pilot studies232\nshowing equilibration within 1.5 seconds.233\nGlutamatergic stimulation consisted of 5 ms rectangular pulses with amplitude Gglu = 1 µM234\nand frequencies ranging from 1–250 Hz (25 logarithmically spaced values). GABAergic stimulation,235\nwhen present, provided continuous background inhibition at three levels: 0 Hz (control), 25 Hz, and236\n50 Hz with GGABA = 0.5 µM.237\n2.8.3 Parameter Space Exploration238\nWe systematically varied βNMDA across 20 logarithmically spaced values (0.01–0.1 ms −1) and stim-239\nulation frequency across 25 values (1–250 Hz), combined with 3 GABAergic conditions, generating240\n1,500 unique parameter combinations (20 × 25 × 3). Each combination was replicated n = 10 times241\nwith different random seeds (using numpy.random with seeds 0–9), totaling 15,000 simulations and242\nyielding 2,942,093 ISI observations after quality control.243\n2.9 Data Analysis Pipeline244\n2.9.1 Spike Detection and Quality Control245\nAction potentials were detected using a dual-threshold algorithm: initial detection at V = 0 mV,246\nconfirmation at V = 20 mV, with a 2 ms absolute refractory period. Inter-spike intervals were247\ncalculated as:248\nISIi = ti+1 − ti (38)\nQuality control procedures excluded: (1) physiologically implausible ISIs ( < 2 ms or > 2000249\nms), (2) simulations with < 10 spikes, and (3) simulations showing numerical instability (voltage250\nexcursions > 200 mV). This resulted in exclusion of 1.2% of the total dataset.251\n2.9.2 Dynamical Analysis Measures252\nShannon Entropy: ISI variability was quantified using 20 equal-width bins based on Sturges’ rule253\nfor the dataset size:254\nH = −\n20X\nk=1\npk log2 pk (39)\nwhere pk represents the probability of the k-th ISI bin. Bin width was determined individually for255\neach parameter combination to ensure adequate sampling.256\n9\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nMaximum Lyapunov Exponents: Estimated using the Wolf et al. (1985) algorithm Wolf257\net al. (1985) with embedding dimension d = 5, time delay τ = 5 ms (determined from first minimum258\nof mutual information), and nearest neighbor radius r = 0.1 mV. Convergence was verified by259\nrequiring < 5% change over the final 25% of the time series. Positive values (λ > 0) indicate chaotic260\ndynamics.261\nMutual Information: Calculated between parameters and ISI patterns using:262\nMI(X; Y ) =\nX\nx,y\np(x, y) log2\np(x, y)\np(x)p(y) (40)\nBias correction was applied using the Miller-Madow estimator Miller (1955). Statistical signifi-263\ncance was assessed using surrogate data generated by 1000 random permutations.264\n2.9.3 Frequency Band Classification265\nISIs were categorized into physiologically relevant bands based on inverse frequency relationships:266\ngamma (7–33 ms, corresponding to 30–143 Hz), beta (33–77 ms, 13–30 Hz), alpha (77–125 ms, 8–13267\nHz), theta (125–250 ms, 4–8 Hz), and delta (250–2000 ms, 0.5–4 Hz). Probability density functions268\nwere estimated using Gaussian kernel density estimation with Scott’s rule for bandwidth selection.269\n2.9.4 Bifurcation and Phase Space Analysis270\nBifurcation diagrams plotted steady-state ISI values against βNMDA after 2000 ms equilibration.271\nLocal maxima and minima were identified using a peak-finding algorithm with minimum prominence272\nof 5% of the dynamic range. Phase portraits were constructed by plotting membrane voltage V (t)273\nversus its numerical derivative dV /dt calculated using central differences.274\n2.9.5 Statistical Analysis and Validation275\nResults are presented as mean ± SEM across n = 10 replications. Statistical significance was276\nassessed using two-way ANOVA ( α = 0 .05) with Bonferroni correction for multiple comparisons.277\nEffect sizes are reported as partial eta-squared (η 2\np). Assumptions were tested using Shapiro-Wilk278\ntests for normality and Levene’s test for homoscedasticity.279\nCross-validation was performed using 5-fold temporal splitting to assess stability of dynamical280\nmeasures. Bootstrap confidence intervals (95%, n = 1000) were calculated for all summary statis-281\ntics. Sensitivity analysis varied key parameters (integration timestep, bin numbers, embedding282\ndimensions) by ±25% to ensure robustness of findings.283\n2.10 Model Limitations284\nThe current model incorporates several simplifications: (1) single-compartment geometry neglect-285\ning dendritic processing, (2) simplified synaptic kinetics without vesicle depletion, (3) absence of286\nnetwork connectivity and population dynamics, and (4) deterministic framework excluding synap-287\ntic noise. These limitations were chosen to maintain computational tractability while preserving288\nessential mechanisms relevant to NMDA receptor kinetics and neuronal excitability.289\n2.11 Reproducibility Statement290\nAll simulation code and analysis scripts are available athttps://github.com/borjkhani/Bifurcation_291\nNMDA. Computational requirements include Python 3.8+ with specified dependencies and approx-292\n10\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nimately 32 GB RAM for full parameter space exploration. Random seed handling ensures repro-293\nducible results, and detailed parameter files enable exact replication of all findings. Raw data and294\nprocessed results are available upon reasonable request in accordance with institutional data sharing295\npolicies.296\n3 Results297\n3.1 Model Validation and Basic Neuronal Response298\nThe computational model reproduced characteristic pyramidal neuron dynamics under controlled299\nstimulation conditions (Figure 3). Glutamatergic inputs (1 µM pulses) and continuous GABAergic300\nbackground activity (0.5 µM) generated distinct synaptic current profiles. NMDA currents exhibited301\nslow kinetics with sustained activation, AMPA currents showed rapid transient responses, and302\nGABA currents provided inhibitory modulation. Regular action potential firing occurred at 30.0 Hz303\nwith realistic membrane potential dynamics ranging from -75 mV to +50 mV. Intracellular calcium304\nconcentration ([Ca2+]i) oscillated between baseline and 0.858µM peaks (mean: 0.256 µM), reflecting305\nNMDA receptor-mediated calcium influx. CaMKII phosphorylation levels at βNMDA = 0 .0520306\nms−1 demonstrated three distinct plasticity regimes: Long-Term Depression (LTD), metaplasticity307\ntransition zone, and Long-Term Potentiation (LTP) regions, with phosphorylation levels plotted on308\nlogarithmic scale over time.309\n3.2 Information-Theoretic Analysis Reveals Optimal NMDA Receptor Modu-310\nlation311\nInformation-theoretic analysis identified optimal NMDA receptor kinetics for neural coding (Figure312\n4). Mutual information between neural responses and stimuli peaked at βNMDA = 0.028 ms−1 with313\nmaximum information transfer of 0.275 bits. Mutual information exhibited frequency-dependent314\ncharacteristics, with highest values in the low-frequency range (0-30 Hz) and rapid decay at higher315\nstimulation frequencies above 50 Hz. Neural response entropy increased with NMDA modulation316\nstrength, reaching a plateau atβNMDA = 0.383 ms−1 with peak entropy of 1.81 bits. Entropy demon-317\nstrated frequency-dependent modulation with a prominent peak at 185 Hz (2.89 bits), followed by318\nsharp decline at higher frequencies. Error bars represent standard deviation across multiple trials.319\n3.3 Bifurcation Analysis Reveals Period-Doubling Routes to Chaos320\nDetailed bifurcation analysis in the low-frequency regime revealed systematic transitions in firing321\ndynamics (Figure 5). Entropy and maximum Lyapunov exponent heatmaps identified chaotic re-322\ngions concentrated in low-frequency, low- βNMDA parameter space. Phase portraits demonstrated323\nprogressive evolution from regular periodic firing ( βNMDA = 0.005 ms −1) through period-doubling324\ncascades to chaotic dynamics ( βNMDA = 0.08 ms−1). Bifurcation diagrams showed period-doubling325\nroutes to chaos with pink regions indicating chaotic parameter ranges. ISI probability density func-326\ntions revealed frequency band evolution from gamma (30-100 Hz) through beta (13-30 Hz) to alpha327\n(8-13 Hz) distributions. CaMKII phosphorylation analysis across different βNMDA values (0.007,328\n0.042, 0.082 ms −1) demonstrated distinct plasticity regimes: LTP region (high phosphorylation),329\ntransition zone, and LTD region (low phosphorylation).330\n11\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n3.4 Frequency-Dependent Neural Dynamics and Phase Space Evolution331\nVoltage traces and phase diagrams demonstrated systematic changes in neural dynamics across332\nstimulation frequencies (Figure 6). At 2 Hz stimulation, neurons exhibited complex firing patterns333\nwith multiple βNMDA values producing distinct voltage trajectories and elaborate phase space struc-334\ntures. At 5 Hz, moderate frequency stimulation showed earlier onset of irregular dynamics with335\nsimplified phase portraits. Higher stimulation frequencies produced progressive stabilization: 15336\nHz demonstrated rapid convergence to regular firing patterns, while 50 Hz stimulation resulted in337\nhighly regular dynamics with compressed phase space trajectories. Phase diagrams revealed sys-338\ntematic evolution from complex multi-loop attractors at low frequencies to simple limit cycles at339\nhigh frequencies. Voltage amplitudes ranged from -75 mV to +50 mV across all conditions, with340\nfiring patterns becoming increasingly predictable as stimulation frequency increased.341\n3.5 Frequency-Dependent Chaos Threshold Erosion342\nBifurcation diagrams revealed systematic erosion of chaos thresholds with increasing stimulation343\nfrequency (Figure 7). At 2 Hz stimulation, chaotic regions (pink shading) occupied 18 parameter344\nranges (2.0% of βNMDA space), with complex bifurcation structures and ISI values ranging from345\n0-500 ms. At 5 Hz, chaotic regions decreased to 22 ranges (2.4% coverage) with ISI compression to346\n0-200 ms. Further frequency increases produced progressive stabilization: 10 Hz exhibited 25 chaotic347\nregions (2.8% coverage) with ISI range 0-80 ms, while 15 Hz showed 20 regions (2.2% coverage) and348\nISI range 0-60 ms. At 50 Hz stimulation, chaos was nearly eliminated with only 5 regions (0.6%349\ncoverage) and tight ISI clustering around 15-18 ms. Bifurcation patterns evolved from complex350\nmulti-branch structures at low frequencies to simple periodic solutions at high frequencies.351\n3.6 Oscillatory Band Evolution and Spectral Consolidation352\nFrequency band analysis revealed systematic spectral evolution across stimulation rates (Figure 8).353\nAt 2 Hz stimulation, ISI distributions showed broad multi-band characteristics spanning gamma354\n(30-100 Hz, blue), beta (13-30 Hz, red), alpha (8-13 Hz, green), theta (4-8 Hz, purple), and delta355\n(0.5-4 Hz, yellow) frequency ranges. Empirical distributions demonstrated peak densities of 0.08 for356\nbeta band and 0.05 for alpha band. At 5 Hz stimulation, spectral content consolidated with reduced357\ndelta component and enhanced gamma representation. Progressive frequency increases produced358\nsystematic band elimination: 10 Hz stimulation showed dominant alpha band concentration (density359\n= 1.0) with minimal beta and theta contributions. At 15 Hz, beta band dominance emerged (density360\n= 0.5) with compressed alpha representation. At 50 Hz stimulation, complete gamma band con-361\ncentration occurred (density = 3.5) with elimination of all slower frequency components. Gaussian362\nmixture model fits confirmed progressive spectral narrowing from multi-component distributions to363\nsingle-component gamma-range concentration.364\n3.7 GABAergic Inhibition Provides Frequency-Selective Stabilization365\nBackground GABAergic inhibition systematically modulated neural bifurcation dynamics across366\ninhibition frequencies from 0-50 Hz (Figure 9). Control conditions (no GABA) exhibited extensive367\nchaotic regions (pink shading) with complex bifurcation structures spanning ISI ranges from 0-100368\nms. GABA inhibition at 2 Hz produced minimal stabilization effects, maintaining similar chaotic369\nparameter coverage. Progressive GABA frequency increases demonstrated systematic stabilization:370\n5 Hz inhibition reduced chaotic regions and simplified bifurcation patterns, while 10 Hz further371\ncompressed chaotic parameter space. At 15 Hz and 20 Hz GABA inhibition, chaotic regions were372\n12\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nsubstantially reduced with ISI ranges constrained to 20-80 ms. Complete chaos elimination occurred373\nat 50 Hz GABA inhibition, producing stable dynamics with ISI clustering around 15-20 ms across the374\nentire βNMDA parameter range. Bifurcation structures evolved from complex multi-branch patterns375\nto simple periodic solutions with increasing GABA frequency.376\n3.8 GABA Modulates CaMKII-Mediated Plasticity States377\nCaMKII phosphorylation analysis revealed systematic GABA-dependent modulation of synaptic378\nplasticity mechanisms (Figure 10). Aggregate CaMKII dynamics across all βNMDA values showed379\nprogressive suppression with increasing GABA frequencies: control conditions (0 Hz) reached maxi-380\nmum phosphorylation levels of 2.5 ×10−23 M, while 50 Hz GABA reduced peak levels to 1.2 ×10−23381\nM. Individual trajectory analysis at βNMDA = 0.007 ms−1 demonstrated GABA-dependent phospho-382\nrylation suppression over 6-second time courses. Plasticity state transitions versus GABA frequency383\nshowed systematic shifts: βNMDA = 0.007 ms −1 decreased from 2.1 to 1.2 ×10−23 M, while higher384\nβNMDA values maintained stable low phosphorylation levels. Phase diagrams revealed exponential385\ndecay relationships between CaMKII levels and βNMDA across GABA conditions. Parameter space386\nanalysis quantified plasticity region redistribution: 0 Hz GABA produced 63% LTD, 27% transition,387\nand 10% LTP regions, while 50 Hz GABA shifted to 83% LTD, 17% transition, and minimal LTP388\ncoverage.389\n4 Discussion390\nOur computational analysis reveals that NMDA receptor kinetics fundamentally control neuronal391\ndynamics through dual pathways, with broad implications across multiple neuropsychiatric con-392\nditions Hansen et al. (2021); Paoletti et al. (2023). While these findings have potential applica-393\ntions to schizophrenia, autism spectrum disorders, Alzheimer’s disease, and chronic pain syndromes394\nGulchina et al. (2024); Zhou and Sheng (2023); Bruining et al. (2024), we focus our detailed dis-395\ncussion on two specific domains where NMDA dysfunction plays particularly well-characterized396\nroles: addiction-related memory formation and visual processing disorders. These applications397\ndemonstrate the translational potential of our quantitative framework for understanding cortical398\nexcitability and developing targeted therapeutic interventions.399\n4.1 Dual Pathways Framework400\nTwo mechanistically distinct routes to firing irregularity emerged from our analysis of 2,942,093 ISI401\nobservations. High-frequency chaos (Pathway 1) occurs under rapid NMDA deactivation (βNMDA >402\n0.06 ms−1) with strong synaptic drive, producing deterministic chaos with compromised information403\nencoding (MI = 0.1847 bits vs. 0.2753 bits optimal) Durstewitz and Gabriel (2007); Toyoizumi and404\nAbbott (2011). Prolonged activation irregularity (Pathway 2) results from slow NMDA deactivation405\n(βNMDA < 0.02 ms −1) creating instability even under weak drive, with sustained calcium influx406\ndriving aberrant CaMKII phosphorylation Zhabotinsky (2000); Abraham (2008).407\nThe identification of an optimal kinetic window ( βNMDA = 0.028 ms −1; MI = 0.2753 bits)408\nrepresents a critical balance point that maximizes spike timing information capacity while avoiding409\npathological states de Ruyter van Steveninck et al. (1997); Schreiber et al. (2009). This framework410\nchallenges traditional views attributing irregular firing solely to synaptic noise or network imbalance,411\ninstead revealing NMDA kinetics as master regulators of cortical excitabilityWang (1999); Ruggiero412\net al. (2024).413\n13\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n4.2 model limitations and cience414\n4.2.1 Pathological Memory Formation415\nThe slow NMDA deactivation pathway (Pathway 2) directly parallels kinetic alterations observed416\nfollowing chronic drug exposure Kalivas(2009); Wolf (2016); Volkow and Boyle(2023). Our findings417\ndemonstrate that prolonged receptor activation maintains elevated CaMKII phosphorylation (8.7 ±418\n0.3 × 10−21 M vs. 1.8 ± 0.1 × 10−23 M for normal kinetics), creating conditions for pathological419\nLTP that differs qualitatively from normal learning-related plasticity L¨ uscher and Malenka(2011);420\nPascoli et al. (2014).421\nThis prolonged activation enables formation of abnormally persistent drug-associated memories422\nthrough sustained calcium influx and aberrant plasticity mechanisms Kauer and Malenka (2007);423\nHearing et al. (2022). Unlike normal memory formation requiring precisely timed calcium tran-424\nsients, drug-induced memories exploit aberrantly extended NMDA activation to create difficult-to-425\nreverse synaptic modifications Borjkhani et al. (2018a,b). The entropy characteristics in this regime426\nsuggest this pathway not only creates pathological plasticity but also disrupts normal information427\nprocessing, potentially explaining cognitive inflexibility observed in addiction Goldstein and Volkow428\n(2011). These findings complement our previous work showing that opioids can induce pathological429\ntheta oscillations associated with addiction memory formation (Borjkhani et al., 2018c), and extend430\nour understanding of how different drugs of abuse alter neural computation through distinct ionic431\nmechanisms (Borjkhani et al., 2022).432\n4.2.2 Therapeutic Targeting Strategies433\nOur findings suggest novel therapeutic approaches targeting NMDA kinetics during memory re-434\nconsolidation Nader et al. (2000); Lee et al. (2006). The chaotic dynamics in Pathway 1 indicate435\nthat pharmacologically accelerating receptor deactivation during memory retrieval could disrupt436\nreconsolidation by degrading the neural code required for memory restabilization Lee et al. (2017).437\nThis approach could selectively target drug memories while preserving normal memory function.438\nThe frequency-dependent stability effects have important implications for cue-induced relapse.439\nThe systematic threshold erosion with increasing frequency (72440\n4.3 Applications to Visual Processing Disorders441\n4.3.1 Retinal Pathophysiology442\nNMDA receptors in retinal ganglion cells are critical for contrast sensitivity, direction selectivity,443\nand light adaptation Manookin et al. (2010); Chen et al. (2016). Our dual-pathway framework444\nreveals mechanistically distinct routes to retinal dysfunction. The prolonged activation pathway445\nmaintains sustained calcium influx directly relevant to excitotoxic mechanisms in glaucoma, where446\nchronic glutamate elevation leads to retinal ganglion cell death Seki and Lipton (2008); Boccuni447\nand Fairless (2022). The quantitative relationship between NMDA kinetics and calcium homeostasis448\nprovides specific targets for neuroprotective interventions Bezprozvanny (2009).449\nThe high-frequency chaos pathway may explain acute retinal injury responses in diabetic retinopa-450\nthy, where rapid metabolic changes alter NMDA kinetics and disrupt normal information encoding451\nBai et al. (2013); Hartwick et al. (2008). The degraded mutual information observed in this regime452\ncould underlie visual processing deficits during acute hyperglycemic episodes.453\n14\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n4.3.2 Cortical Visual Processing454\nThe systematic oscillatory pattern shifts have direct relevance for cortical visual processing, where455\ngamma oscillations mediate feature binding and attention while alpha rhythms control spatial atten-456\ntion and predictive coding Jensen and Mazaheri (2010); Singer (1999). Our frequency-dependent457\nanalysis reveals that NMDA kinetics fundamentally control the balance between these computa-458\ntional modes Fox and Daw (1992); Nowak et al. (1997).459\nVisual processing disorders may result from NMDA kinetic alterations disrupting normal oscil-460\nlatory balance, potentially explaining visual attention deficits in conditions like amblyopia where461\nNMDA function changes during critical developmental periods Hensch (2005); Takesian and Hensch462\n(2022); Fagiolini et al. (2024). The optimal kinetic window we identify may represent evolutionary463\noptimization for experience-dependent visual development Bavelier et al. (2010); Meredith et al.464\n(2023).465\n4.4 GABAergic Modulation and Circuit Stabilization466\nGABAergic inhibition provided frequency-selective stabilization, expanding stable parameter space467\nby 34.2468\nFor addiction treatment, GABAergic modulation could prevent transition to chaotic regimes469\nduring cue exposure while maintaining normal memory encoding capacity. In visual disorders,470\nfrequency-selective GABA enhancement could restore optimal oscillatory balance required for visual471\nattention and processing Whittington et al. (2000); Brunel and Hakim (1999). The complete chaos472\nelimination at 50 Hz GABA demonstrates the therapeutic potential of precisely timed inhibitory473\nmodulation Yizhar et al. (2011).474\n4.5 Clinical Translation and Precision Medicine475\nThe parameter-dependent effects we identify suggest that therapeutic interventions should be tai-476\nlored to individual NMDA dysfunction patterns rather than employing broad-spectrum approaches477\nGlasgow et al. (2022); Traynelis et al. (2010). For patients showing evidence of slow NMDA kinet-478\nics (Pathway 2), interventions that accelerate receptor deactivation during memory retrieval may479\nbe beneficial for addiction treatment. Conversely, individuals with rapid kinetics and high neural480\nvariability may require stabilization approaches enhancing GABAergic inhibition.481\nOur computational framework suggests several neurophysiological biomarkers for assessing cir-482\ncuit states and guiding personalized interventions: EEG-derived entropy measures could indicate483\nproximity to chaotic regimes Ellner and Turchin (1995); Wolf et al. (1985), gamma/beta power484\nratios could reflect underlying NMDA kinetic states Uhlhaas and Singer (2010); Donner and Siegel485\n(2011), and stimulus-response mutual information could quantify encoding process integrity and486\ntreatment efficacy Goebel et al. (2005); Reeke and Coop (2004).487\nRather than broad NMDA antagonism that can impair normal cognitive function, our find-488\nings suggest kinetically-specific approaches Javitt (2007); Lisman et al. (2008). Positive allosteric489\nmodulators that normalize deactivation rates could restore optimal information processing while490\npreserving beneficial NMDA-dependent functions Lutzu and Castillo (2021); Carles et al. (2024).491\nThe frequency-selective effects of GABAergic modulation indicate that targeted stimulation proto-492\ncols could provide therapeutic benefits for disorders involving NMDA dysfunction.493\n15\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n4.6 Model Limitations and Future Directions494\nSeveral limitations should be acknowledged when interpreting our findings. Our single-cell approach495\ncannot capture network-level phenomena like synchronization and large-scale oscillations crucial for496\naddiction circuits and visual processing networks Cabral et al. (2014); Anticevic et al. (2012).497\nNMDA receptor subtypes and kinetic properties vary across brain regions, requiring region-specific498\nparameter adjustments for clinical applications Monyer et al. (1992); Standaert et al. (1999); Sheng499\net al. (1994). Different neuronal subtypes exhibit distinct NMDA-dependent behaviors that our500\ngeneralized model does not capture Spruston (2008); Freund and Buzs´ aki(1996).501\nDespite these limitations, the quantitative relationships we establish between receptor kinetics502\nand neural dynamics provide a foundational framework for understanding NMDA-related pathology.503\nFuture experimental validation should include dynamic clamp experiments manipulating NMDA504\nkinetics directly Harsch and Robinson (2000); Vargas-Caballero and Robinson (2004), optogenetic505\napproaches for selective in vivo modification Boyden (2011), and high-density electrophysiology506\nmeasuring oscillatory changes in behaving animals during addiction-related and visual processing507\ntasks.508\nNetwork-level extensions should incorporate recurrent connectivity patterns Destexhe and Se-509\njnowski (2009); Izhikevich (2006), multiple cell types with distinct NMDA properties, and region-510\nspecific circuit architectures relevant to addiction and visual processing Deco et al. (2008, 2023).511\nThese developments will advance understanding of NMDA receptors as master regulators of corti-512\ncal computation and therapeutic targets for diverse brain disorders, with particular relevance for513\ndeveloping precision medicine approaches to addiction treatment and visual processing disorder514\ninterventions.515\n5 Conclusion516\nOur computational investigation reveals NMDA receptor kinetics as fundamental controllers of neu-517\nronal excitability and synaptic plasticity through dual mechanistic pathways. Analysis of 2,942,093518\ninter-spike intervals identified two distinct routes to firing irregularity: high-frequency chaos under519\nrapid deactivation and prolonged activation irregularity under slow deactivation, with an optimal520\nkinetic window at βNMDA = 0.028 ms−1 maximizing information encoding.521\nThese findings provide mechanistic insights into addiction-related memory formation, where522\nprolonged NMDA activation creates conditions for pathological plasticity, and visual processing523\ndisorders, where altered kinetics disrupt retinal function and cortical oscillatory balance. GABAer-524\ngic inhibition offers frequency-selective stabilization, expanding stable parameter space by 34.2%525\nwhile preserving beneficial gamma rhythms.526\nThe quantitative relationships between molecular receptor properties and network dynamics527\nestablish a framework for precision medicine approaches targeting kinetically-specific interventions.528\nAs tools for measuring and manipulating NMDA function advance, these insights will prove crucial529\nfor translating molecular discoveries into effective clinical interventions across NMDA-related brain530\ndisorders.531\n6 Data and Code Availability532\nAll simulation code, analysis scripts, and datasets are publicly available at: https://github.com/533\nborjkhani/Bifurcation_NMDA. The repository includes fully documented simulation code, param-534\neter files, analysis scripts, raw and processed data files, figure generation scripts, and computational535\n16\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nenvironment setup instructions. Simulations were performed using Python 3.8 with fixed random536\nseeds (seed = 42) for complete reproducibility.537\nAcknowledgments538\nWe thank Prof. Maciej Wojtkowski for helpful discussions and support. We acknowledge ICTER –539\nInternational Centre for Translational Eye Research for providing computational resources.540\nAuthor Contributions541\nM.B. conceived the study, developed the computational model, performed simulations, and wrote542\nthe manuscript. H.B. contributed to the development of the computational model and performed543\nsimulations and statistical analysis. M.A.S. provided expertise in bifurcation analysis. F.B. super-544\nvised the project and provided critical feedback. M.J. contributed to model validation. All authors545\nreviewed and approved the final manuscript.546\nCompeting Interests547\nThe authors declare no competing financial or non-financial interests.548\nReferences549\nWickliffe C Abraham. Metaplasticity: tuning synapses and networks for plasticity. 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Trends in Neuro-802\nsciences, 39(3):136–145, 2016.803\n23\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nA Statistical Analysis Methods804\nA.1 Dataset Characteristics805\nThe simulation generated 2,942,093 ISI observations across 1,961 parameter combinations. Quality806\ncontrol procedures included ISI filtering (2-2000 ms), outlier detection, and convergence verification.807\nA.2 Core Statistical Procedures808\nTwo-way ANOVA examined main effects of βN M DA categories and stimulation frequency on neu-809\nronal dynamics, with post-hoc comparisons using independent samples t-tests and Cohen’s d effect810\nsizes.811\nA.3 Dynamical Analysis Methods812\nShannon entropy: H = − P20\nk=1 pk log2 pk Maximum Lyapunov exponents: Wolf et al. algo-813\nrithm with embedding dimension d=5, time delay τ=5 ms Mutual information: M I(X; Y ) =814\nP\nx,y p(x, y) log2\np(x,y)\np(x)p(y)815\nA.4 Software and Reproducibility816\nAnalyses used Python 3.8 with NumPy 1.21.0, SciPy 1.7.0, scikit-learn 1.0.0. Code available with817\nfixed random seeds (seed=42) for reproducibility.818\n24\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nModel Development & Parameter Space\nDynamical Analysis Methods\nKey Findings: Dual Pathways F ramework\nClinical T ranslation & Applications\nHodgkin–Huxley\nPyramidal\nNeuron\n• Na+, K +,\nCa2+ currents\n• NMDA,\nAMP A, GABA\n• CaMKII phos-\nphorylation\nβNMDA\n0.01–0.1 ms−1\n20 log values\nStimulation\nFreq\n1–250 Hz\n25 values\nComputational\nSimulation\n• Runge–Kutta 4th\n• ∆t = 0.05 ms\n• 1 961 combos\n• 2 942 093 ISI\nShannon\nEntropy\nH =\n− ∑pk log2pk\nMaximum\nLyapunov\nExp.\nChaos detection\nMutual\nInformation\nInfo encoding\nBifurcation\nAnalysis\nPeriod-doubling\nFrequency\nBand\nAnalysis\nγ,β,α,θ,δ\nStatistical\nAnalysis\nTwo-way ANOVA\np <0.001\nPathway 1:\nHigh-Freq\nChaos\nβNMDA>\n0.06 ms−1\nEntropy: 1.441 bits\nCompromised\nencoding\nPathway 2:\nProlonged\nActivation\nβNMDA<\n0.02 ms−1\nEntropy: 1.347 bits\nSustained Ca2+\nOptimal\nWindow\nβNMDA= 0.028\nMI = 0.2753 bits\nMax information\nGABA Effects\n34.2% ex-\npansion\n43% chaos reduction\nFrequency-selective\nAddiction\nMemory\nInterventions\n• Pathological L TP\n• Reconsolidation\n• Kinetic targets\nVisual Pro-\ncessing\nDisorders\n• Retinal\ndysfunction\n• Oscilla-\ntion restore\n• γ/αbalance\nTherapeutic\nStrategies\n• Precision\nmedicine\n• Kinetic\nmodulators\n• Biomarkers\nParameter DefinitionAnalysis PipelineMechanistic Insights\nData GenerationPattern RecognitionClinical Translation\nMethodology Flow: Direct\nCross\nScale of Analysis:\n• 1 961 combinations\n• 2 942 093 ISI obs.\n• 20βNMDAvalues\n• 25 frequencies\n• 10-s runs\n• Replicates\nFigure 1: Comprehensive computational workflow for investigating NMDA receptor ki-\nnetics and neuronal dynamics. The study employed a systematic four-stage approach: (1)\nModel Development & Parameter Space - Implementation of a detailed Hodgkin-Huxley pyramidal\nneuron model with systematic exploration of NMDA receptor closing rates (βN M DA: 0.01-0.1 ms −1,\n20 values) and glutamatergic stimulation frequencies (1-250 Hz, 25 values). GABAergic modulation\nwas tested at three levels (0, 25, 50 Hz), generating 1,500 parameter combinations (500 base ×\n3 GABA conditions). (2) Dynamical Analysis Methods - Application of multiple complementary\ntechniques including Shannon entropy analysis, maximum Lyapunov exponent calculation, mutual\ninformation quantification, bifurcation analysis, frequency band decomposition, and statistical vali-\ndation. (3) Key Findings: Dual Pathways Framework - Identification of two distinct routes to firing\nirregularity (Pathway 1: high-frequency chaos; Pathway 2: prolonged activation) and an optimal\nkinetic window for information encoding, alongside GABAergic modulation effects. (4) Clinical\nTranslation & Applications - Direct relevance to addiction memory interventions, visual processing\ndisorders, and therapeutic strategy development. The analysis encompassed 2,942,093 inter-spike\ninterval observations across multiple computational replicates. Solid arrows indicate direct analyti-\ncal flow; dotted arrows represent cross-validation and mechanistic connections between findings and\napplications.\n25\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n𝑰 𝑵𝒂 𝑰 𝑪\n𝑰𝑵𝒂𝑷 𝑰𝑲𝒅𝒓\n𝑰 𝑨\n𝑰 𝑴\n𝑰𝑪𝒂 𝑰 𝒔𝑨𝑯𝑷\n𝑰𝑵𝑴𝑫𝑨 𝑰𝑨𝑴𝑷𝑨\n𝑰𝑮𝑨𝑩𝑨\n𝑰𝑮𝑨𝑩𝑨\n𝑮𝑨𝑩𝑨\n𝑮𝒍𝒖𝒕𝒂𝒎𝒂𝒕𝒆\nCa2+\nK+\nNa+\nFigure 2: Synaptic inputs and ionic currents in the modeled neuron. This schematic\nillustrates the synaptic and intrinsic ionic currents included in the model. The total synaptic\ncurrent consists of AMPA, NMDA, and GABAergic components, where AMPA and NMDA mediate\nexcitatory transmission, with NMDA exhibiting a voltage-dependent Mg2+ block, while GABAergic\ncurrents provide inhibition. The intrinsic ionic currents include fast and persistent sodium ( INa,\nINaP), various potassium currents regulating repolarization and adaptation ( IKdr, IA, IM), calcium-\nmediated currents ( ICa, IC), and slow afterhyperpolarization and leak currents ( IsAHP, IL). These\ncurrents collectively shape neuronal excitability, synaptic integration, and firing dynamics.\n26\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n0.0\n0.5\n1.0\nGlutamate\nInput (?M)\nGlutamatergic Synaptic Input\n0.0\n0.5\n1.0\nGABA\n(?M)\nGABAergic Input\n?30\n?20\n?10\n0\nINMDA\n(nA)\nNMDA Current\n?20\n?10\n0\nIAMPA\n(nA)\nAMPA Current\n0\n20\n40\nIGABA\n(nA)\nGABA Current\n?50\n0\n50\nMembrane Potential\n(mV)\nSpike Rate: 30.0 Hz\nAction Potentials and Neural Activity\n0.00\n0.25\n0.50\n0.75\n/uni0000003e/uni00000026/uni00000044/uni000000f0/uni00000040i\n(?M)\nMean: 0.256 ?M\nPeak: 0.858 ?M\nIntracellular Calcium Dynamics\n5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4\nTime (s)\n/uni00000014/uni00000013/uni00000015/uni00000016\n/uni00000014/uni00000013/uni00000015/uni00000015\n/uni00000014/uni00000013/uni00000015/uni00000014\n/uni00000014/uni00000013/uni00000015/uni00000013\nPhosphorylated CaMKII\n(?M, log scale)\nLong-Term\nDepression\nMetaplasticity\nTransition\nLong-Term\nPotentiation\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\nCaMKII Phosphorylation\nLTD Region\nTransition\nLTP Region\nNeural Dynamics and Synaptic Plasticity: CaMKII-Mediated Learning\nA\nB\nC\nD\nE\nFigure 3: Neural dynamics and synaptic plasticity: CaMKII-mediated learning. (A)\nSynaptic input patterns: glutamatergic stimulation (left, 1 µM pulses) and GABAergic inhibitory\ninput (right, continuous background activity). (B) Synaptic currents showing NMDA-mediated\n(INMDA), AMPA-mediated (I AMPA), and GABA-mediated ( IGABA) responses during stimulation.\n(C) Action potential generation and neural activity with spike rate of 30.0 Hz, demonstrating\nrealistic membrane potential dynamics. (D) Intracellular calcium dynamics ([Ca 2+]i) with mean\nconcentration of 0.256 µM and peak values reaching 0.858 µM, reflecting NMDA receptor-mediated\ncalcium influx. (E) CaMKII phosphorylation dynamics ( βNMDA = 0.0520) showing distinct plastic-\nity regimes: Long-Term Depression (LTD) region, metaplasticity transition zone, and Long-Term\nPotentiation (LTP) region, with CaMKII phosphorylation levels plotted on logarithmic scale over\ntime.\n27\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n0.0 0.2 0.4 0.6 0.8\nNMDA\n0.235\n0.240\n0.245\n0.250\n0.255\n0.260\n0.265\n0.270\n0.275Mutual Information (bits)\nPeak: 0.028\n0.275 bits\n/uni00000024/uni0000000c/uni00000003/uni00000030/uni00000058/uni00000057/uni00000058/uni00000044/uni0000004f/uni00000003/uni0000002c/uni00000051/uni00000049/uni00000052/uni00000055/uni00000050/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000059/uni00000056/uni00000003NMDA\n0 50 100 150 200\nStimulation Frequency (Hz)\n0.0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6Mutual Information (bits)\nB) Mutual Information vs Frequency\nHigh MI region\n0.0 0.2 0.4 0.6 0.8\nNMDA\n0.5\n1.0\n1.5\n2.0\n2.5Entropy (bits)\nPeak: ?=0.383\n1.81 bits\n/uni00000026/uni0000000c/uni00000003/uni00000028/uni00000051/uni00000057/uni00000055/uni00000052/uni00000053/uni0000005c/uni00000003/uni00000059/uni00000056/uni00000003NMDA\n0 50 100 150 200\nStimulation Frequency (Hz)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0Entropy (bits)\nPeak: 185 Hz\n2.89 bits\nD) Entropy vs Stimulation Frequency\nFigure 4: Information-theoretic analysis reveals optimal NMDA receptor modulation\nand frequency-dependent neural coding. (A) Mutual information between neural responses\nand stimuli as a function of NMDA receptor modulation strength (β NMDA). Peak information\ntransfer occurs at βNMDA = 0.028, corresponding to 0.275 bits. (B) Mutual information varies with\nstimulation frequency, showing maximal information content in the low-frequency range (0-30 Hz,\nshaded region) with rapid decay at higher frequencies. (C) Neural response entropy increases with\nNMDA modulation strength, reaching a plateau around βNMDA = 0.4 (peak: 1.81 bits at βNMDA =\n0.383). Error bars represent standard deviation across trials. (D) Entropy exhibits frequency-\ndependent modulation with a prominent peak at 185 Hz (2.89 bits), followed by a sharp decline.\nError bars represent standard deviation. The analysis demonstrates that moderate NMDA receptor\nmodulation optimizes information transfer while maintaining response diversity, with distinct coding\nregimes for low-frequency information transmission and high-frequency response entropy.\n28\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nA B\nC\nD\nE\nF\nFigure 5: Bifurcation analysis and oscillatory dynamics. (A-B) Entropy and MLE heatmaps\nfor low-frequency, low-βNMDA parameter space. (C) Phase portraits showing transition from regular\nto chaotic firing. (D) Bifurcation diagram revealing period-doubling route to chaos (red regions\nindicate chaotic dynamics). (E) ISI probability density functions categorized by frequency bands.\n(F) CaMKII phosphorylation levels across plasticity regimes: LTP (orange), transition (green), LTD\n(blue).\n29\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\nA B\nC D\nFigure 6: Neural dynamics across stimulation frequencies. Voltage traces and phase diagrams\nshowing progressive destabilization from (A) 2 Hz stable dynamics with complex bifurcations, (B)\n5 Hz moderate frequency showing earlier chaos onset, (C) 15 Hz rapid destabilization, to (D) 50 Hz\nimmediate chaotic dynamics with highly irregular patterns.\n30\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000015/uni00000013/uni00000011/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000019/uni00000013/uni00000011/uni00000013/uni0000001b\n/uni00000013\n/uni00000014/uni00000013/uni00000013\n/uni00000015/uni00000013/uni00000013\n/uni00000016/uni00000013/uni00000013\n/uni00000017/uni00000013/uni00000013\n/uni00000018/uni00000013/uni00000013/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000024 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/uni00000014/uni00000013/uni00000003/uni0000002b/uni0000005d/uni00000003/uni00000036/uni00000057/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000016/uni0000001b/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000015/uni00000013/uni00000011/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000019/uni00000013/uni00000011/uni00000013/uni0000001b\n/uni00000013\n/uni00000014/uni00000013\n/uni00000015/uni00000013\n/uni00000016/uni00000013\n/uni00000017/uni00000013\n/uni00000018/uni00000013\n/uni00000019/uni00000013/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000027 /uni00000014/uni00000018/uni00000003/uni0000002b/uni0000005d/uni00000003/uni00000036/uni00000057/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000016/uni0000001b/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000015/uni00000013/uni00000011/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000019/uni00000013/uni00000011/uni00000013/uni0000001b\nNMDA\n/uni00000015/uni00000011/uni00000018\n/uni00000018/uni00000011/uni00000013\n/uni0000001a/uni00000011/uni00000018\n/uni00000014/uni00000013/uni00000011/uni00000013\n/uni00000014/uni00000015/uni00000011/uni00000018\n/uni00000014/uni00000018/uni00000011/uni00000013\n/uni00000014/uni0000001a/uni00000011/uni00000018/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000028 /uni00000018/uni00000013/uni00000003/uni0000002b/uni0000005d/uni00000003/uni00000036/uni00000057/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni0000000b/uni00000003/uni00000020/uni00000003/uni00000013/uni00000011/uni00000016/uni0000001b/uni0000000c\nFigure 7: Frequency-dependent bifurcation landscapes and chaos onset thresholds. (A-\nE) Bifurcation diagrams showing inter-spike interval distributions versus βNMDA across stimulation\nfrequencies from 2 Hz to 50 Hz. Pink shading indicates chaotic regions. Progressive frequency\nincreases demonstrate systematic erosion of stability thresholds: (A) 2 Hz exhibits complex bifur-\ncations with chaos onset at βNMDA ≈ 0.012 ms −1, (B) 5 Hz shows earlier destabilization, (C) 10\nHz displays limited chaotic windows, (D) 15 Hz demonstrates compressed parameter ranges, and\n(E) 50 Hz maintains remarkable stability with tight gamma-frequency ISI clustering (15-18 ms) and\nminimal chaotic behavior.\n31\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n/uni00000013/uni00000014/uni00000013/uni00000013/uni00000015/uni00000013/uni00000013/uni00000016/uni00000013/uni00000013/uni00000017/uni00000013/uni00000013/uni00000018/uni00000013/uni00000013\n/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000014\n/uni00000013/uni00000011/uni00000013/uni00000015\n/uni00000013/uni00000011/uni00000013/uni00000016\n/uni00000013/uni00000011/uni00000013/uni00000017\n/uni00000013/uni00000011/uni00000013/uni00000018\n/uni00000013/uni00000011/uni00000013/uni00000019\n/uni00000013/uni00000011/uni00000013/uni0000001a\n/uni00000013/uni00000011/uni00000013/uni0000001b/uni00000033/uni00000055/uni00000052/uni00000045/uni00000044/uni00000045/uni0000004c/uni0000004f/uni0000004c/uni00000057/uni0000005c/uni00000003/uni00000027/uni00000048/uni00000051/uni00000056/uni0000004c/uni00000057/uni0000005c\n/uni00000024/uni00000015/uni00000003/uni0000002b/uni0000005d/uni00000003/uni00000036/uni00000057/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051\n/uni00000003/uni0000000b/uni00000016/uni00000013/uni00000010/uni00000014/uni00000013/uni00000013/uni00000003/uni0000002b/uni0000005d/uni0000000c\n/uni00000003/uni0000000b/uni00000014/uni00000016/uni00000010/uni00000016/uni00000013/uni00000003/uni0000002b/uni0000005d/uni0000000c\n/uni00000003/uni0000000b/uni0000001b/uni00000010/uni00000014/uni00000016/uni00000003/uni0000002b/uni0000005d/uni0000000c\n/uni00000003/uni0000000b/uni00000017/uni00000010/uni0000001b/uni00000003/uni0000002b/uni0000005d/uni0000000c\n/uni00000003/uni0000000b/uni00000013/uni00000011/uni00000018/uni00000010/uni00000017/uni00000003/uni0000002b/uni0000005d/uni0000000c\n/uni00000013/uni00000014/uni00000013/uni00000013/uni00000015/uni00000013/uni00000013/uni00000016/uni00000013/uni00000013/uni00000017/uni00000013/uni00000013/uni00000018/uni00000013/uni00000013\n/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000015\n/uni00000013/uni00000011/uni00000013/uni00000017\n/uni00000013/uni00000011/uni00000013/uni00000019\n/uni00000013/uni00000011/uni00000013/uni0000001b\n/uni00000013/uni00000011/uni00000014/uni00000013/uni00000033/uni00000055/uni00000052/uni00000045/uni00000044/uni00000045/uni0000004c/uni0000004f/uni0000004c/uni00000057/uni0000005c/uni00000003/uni00000027/uni00000048/uni00000051/uni00000056/uni0000004c/uni00000057/uni0000005c\n/uni0000002a/uni00000044/uni00000058/uni00000056/uni00000056/uni0000004c/uni00000044/uni00000051/uni00000003/uni00000029/uni0000004c/uni00000057/uni00000056\n/uni00000013/uni00000018/uni00000013/uni00000014/uni00000013/uni00000013/uni00000014/uni00000018/uni00000013/uni00000015/uni00000013/uni00000013\n/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000015/uni00000018\n/uni00000013/uni00000011/uni00000013/uni00000018/uni00000013\n/uni00000013/uni00000011/uni00000013/uni0000001a/uni00000018\n/uni00000013/uni00000011/uni00000014/uni00000013/uni00000013\n/uni00000013/uni00000011/uni00000014/uni00000015/uni00000018\n/uni00000013/uni00000011/uni00000014/uni00000018/uni00000013\n/uni00000013/uni00000011/uni00000014/uni0000001a/uni00000018\n/uni00000013/uni00000011/uni00000015/uni00000013/uni00000013/uni00000027/uni00000048/uni00000051/uni00000056/uni0000004c/uni00000057/uni0000005c\n/uni00000025/uni00000018/uni00000003/uni0000002b/uni0000005d/uni00000003/uni00000036/uni00000057/uni0000004c/uni00000050/uni00000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051\n/uni00000013/uni00000018/uni00000013/uni00000014/uni00000013/uni00000013/uni00000014/uni00000018/uni00000013/uni00000015/uni00000013/uni00000013\n/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni0000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000058/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051\n/uni00000013/uni00000014/uni00000013/uni00000015/uni00000013/uni00000016/uni00000013/uni00000017/uni00000013\n/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013\n/uni00000013/uni00000011/uni00000013/uni00000015\n/uni00000013/uni00000011/uni00000013/uni00000017\n/uni00000013/uni00000011/uni00000013/uni00000019\n/uni00000013/uni00000011/uni00000013/uni0000001b\n/uni00000013/uni00000011/uni00000014/uni00000013\n/uni0000002a/uni00000044/uni00000058/uni00000056/uni00000056/uni0000004c/uni00000044/uni00000051/uni00000003/uni00000029/uni0000004c/uni00000057/uni00000056\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\n/uni00000028/uni00000050/uni00000053/uni0000004c/uni00000055/uni0000004c/uni00000046/uni00000044/uni0000004f\n/uni00000027/uni0000004c/uni00000056/uni00000057/uni00000055/uni0000004c/uni00000045/uni00000058/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000056\n/uni0000002a/uni00000044/uni00000058/uni00000056/uni00000056/uni0000004c/uni00000044/uni00000051\n/uni00000029/uni0000004c/uni00000057/uni00000056\nFigure 8: Frequency band analysis of neural oscillations across stimulation rates. (A-E)\nTop panels show empirical probability density distributions of inter-spike intervals decomposed into\nphysiological frequency bands: gamma (30-100 Hz, blue), beta (13-30 Hz, red), alpha (8-13 Hz,\ngreen), theta (4-8 Hz, purple), and delta (0.5-4 Hz, yellow) across stimulation frequencies from 2\nHz to 50 Hz. Bottom panels display corresponding Gaussian mixture model fits. Progressive stim-\nulation rate increases produce systematic spectral consolidation: (A) 2 Hz shows broad multi-band\ndistributions, (B-D) intermediate frequencies (5-15 Hz) exhibit gradual gamma-band dominance,\nand (E) 50 Hz stimulation results in tight gamma-range concentration with minimal spectral diver-\nsity.\n32\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint 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/uni0000002a/uni00000024/uni00000025/uni00000024/uni00000003/uni0000002c/uni00000051/uni0000004b/uni0000004c/uni00000045/uni0000004c/uni00000057/uni0000004c/uni00000052/uni00000051/uni0000001d/uni00000003/uni00000015/uni00000013/uni00000003/uni0000002b/uni0000005d\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000014/uni00000013/uni00000011/uni00000013/uni00000015/uni00000013/uni00000011/uni00000013/uni00000016/uni00000013/uni00000011/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000018/uni00000013/uni00000011/uni00000013/uni00000019/uni00000013/uni00000011/uni00000013/uni0000001a/uni00000013/uni00000011/uni00000013/uni0000001b/uni00000013/uni00000011/uni00000013/uni0000001c\nNMDA/uni00000003/uni0000000b/uni00000050/uni000000561/uni0000000c\n/uni00000013\n/uni00000015/uni00000013\n/uni00000017/uni00000013\n/uni00000019/uni00000013\n/uni0000001b/uni00000013\n/uni00000014/uni00000013/uni00000013/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000010/uni00000036/uni00000053/uni0000004c/uni0000004e/uni00000048/uni00000003/uni0000002c/uni00000051/uni00000057/uni00000048/uni00000055/uni00000059/uni00000044/uni0000004f/uni00000003/uni0000000b/uni00000050/uni00000056/uni0000000c\n/uni00000026/uni0000004b/uni00000044/uni00000052/uni00000057/uni0000004c/uni00000046/uni00000003/uni00000024/uni00000055/uni00000048/uni00000044\n/uni00000018/uni00000013/uni00000003/uni0000002b/uni0000005d\n/uni0000002a /uni0000002a/uni00000024/uni00000025/uni00000024/uni00000003/uni0000002c/uni00000051/uni0000004b/uni0000004c/uni00000045/uni0000004c/uni00000057/uni0000004c/uni00000052/uni00000051/uni0000001d/uni00000003/uni00000018/uni00000013/uni00000003/uni0000002b/uni0000005d\n/uni0000002a/uni00000024/uni00000025/uni00000024/uni00000010/uni00000030/uni00000048/uni00000047/uni0000004c/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000031/uni00000048/uni00000058/uni00000055/uni00000044/uni0000004f/uni00000003/uni00000025/uni0000004c/uni00000049/uni00000058/uni00000055/uni00000046/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000027/uni0000005c/uni00000051/uni00000044/uni00000050/uni0000004c/uni00000046/uni00000056\nFigure 9: GABAergic inhibition modulates neural bifurcation dynamics and synaptic\nplasticity. (A-G) Bifurcation diagrams showing inter-spike interval patterns versus βNMDA under\nincreasing GABA inhibition frequencies (0-50 Hz). Pink shading indicates chaotic regions. GABA\nprogressively stabilizes dynamics, with 50 Hz completely eliminating chaos. (H) CaMKII phospho-\nrylation dynamics across GABA conditions, showing plasticity state transitions from LTP (green)\nthrough transition zones to LTD (pink) regions. GABA inhibition systematically reduces CaMKII\nlevels and shifts plasticity thresholds, demonstrating frequency-dependent modulation of synaptic\nplasticity mechanisms.\n33\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint \n\n0 1 2 3 4 5 6\nTime (s)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Phospho-CaMKII ( M)\n1e 23\nA A. CaMKII Dynamics with Uncertainty\n(All NMDA Values)\nGABA 0 Hz\nGABA 20 Hz\nGABA 50 Hz\n0 1 2 3 4 5 6\nTime (s)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Phospho-CaMKII ( M)\n1e 23\nB B. Individual Trajectories\n( NMDA  0.007)\nGABA 0 Hz\nGABA 20 Hz\nGABA 50 Hz\n0 10 20 30 40 50\nGABA Frequency (Hz)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Final CaMKII Level ( M)\n1e 23\nC C. Plasticity State Transitions\nvs GABA Frequency\nNMDA = 0.007\nNMDA = 0.020\nNMDA = 0.050\nNMDA = 0.080\n0.00 0.02 0.04 0.06 0.08\nNMDA\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Final CaMKII Level ( M)\n1e 23\nD D. Plasticity Phase Diagram\\nvs NMDA\nGABA 0 Hz\nGABA 2 Hz\nGABA 5 Hz\nGABA 10 Hz\nGABA 15 Hz\nGABA 20 Hz\nGABA 50 Hz\n0.00 0.02 0.04 0.06 0.08\nNMDA\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Final CaMKII\n1e 23\nE1 E1. GABA 0 Hz\n0.00 0.02 0.04 0.06 0.08\nNMDA\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Final CaMKII\n1e 23\nE2 E2. GABA 20 Hz\n0.00 0.02 0.04 0.06 0.08\nNMDA\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5Final CaMKII\n1e 23\nE3 E3. GABA 50 Hz\n0 Hz 20 Hz 50 Hz\nGABA Frequency\n0\n20\n40\n60\n80\n100\n120\nParameter Space\nCoverage (%)63%\n27%\n10%\n80%\n17%\n83%\n17%\n 3 regions\n  3 regions\n  2 regions\nF F. Plasticity Region Distribution\nAcross NMDA Parameter Space\nLTD Region\nTransition Region\nLTP Region\nGABA-Mediated Modulation of CaMKII Phosphorylation and Synaptic Plasticity\nComprehensive Analysis of LTP/LTD State Transitions\nFigure 10: GABA-mediated modulation of CaMKII phosphorylation and synaptic plas-\nticity. (A) CaMKII dynamics across all βNMDA values showing uncertainty bands for different\nGABA frequencies (0, 20, 50 Hz). (B) Individual CaMKII trajectories at βNMDA = 0.007 ms −1\ndemonstrating GABA-dependent suppression. (C) Plasticity state transitions versus GABA fre-\nquency for four βNMDA values, with horizontal dashed lines indicating LTP/LTD thresholds. (D)\nPhase diagram showing CaMKII levels versus βNMDA across GABA frequencies. (E1-E3) Detailed\nCaMKII curves for 0, 20, and 50 Hz GABA with plasticity region color coding. (F) Parameter\nspace distribution showing progressive shift from LTP-dominant (63%) to LTD-dominant (83%)\nstates with increasing GABA inhibition.\n34\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted September 11, 2025. ; https://doi.org/10.1101/2025.09.06.674628doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}