A unified model of cortico-hippocampal interactions through neural field theory

A unified model of cortico-hippocampal interactions through neural field theory | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results A unified model of cortico-hippocampal interactions through neural field theory View ORCID Profile Richa Phogat , Anna Behler , Saurabh Sonkusare , View ORCID Profile James C. Pang , Nikitas Koussis , James A. Roberts , Jordan DeKraker , View ORCID Profile James M. Shine , Alex Fornito , P. A. Robinson , Michael Breakspear doi: https://doi.org/10.1101/2025.09.07.674721 Richa Phogat 1 School of Psychological Sciences, College of Engineering, Science, and the Environment, University of Newcastle , Callaghan, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Richa Phogat Anna Behler 1 School of Psychological Sciences, College of Engineering, Science, and the Environment, University of Newcastle , Callaghan, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site Saurabh Sonkusare 1 School of Psychological Sciences, College of Engineering, Science, and the Environment, University of Newcastle , Callaghan, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site James C. Pang 2 School of Psychological Sciences, Turner Institute for Brain and Mental Health, and Monash Biomedical Imaging, Monash University , Clayton, Victoria, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for James C. Pang Nikitas Koussis 3 Mark Hughes Foundation Centre for Brain Cancer Research, College of Health, Medicine and Wellbeing, University of Newcastle , Callaghan, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site James A. Roberts 4 Brain Modelling Group, QIMR Berghofer , Herston, Brisbane, Queensland, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site Jordan DeKraker 5 Montreal Neurological Institute and Hospital, McGill University , Montreal, Canada Find this author on Google Scholar Find this author on PubMed Search for this author on this site James M. Shine 6 School of Medical Sciences, Faculty of Medicine and Health, The University of Sydney , Sydney, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for James M. Shine Alex Fornito 2 School of Psychological Sciences, Turner Institute for Brain and Mental Health, and Monash Biomedical Imaging, Monash University , Clayton, Victoria, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site P. A. Robinson 7 School of Physics, University of Sydney , Camperdown, New South Wales, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site Michael Breakspear 1 School of Psychological Sciences, College of Engineering, Science, and the Environment, University of Newcastle , Callaghan, New South Wales, Australia 8 School of Medicine and Public Health, College of Medicine, Health and Wellbeing, University of Newcastle , Newcastle, NSW, Australia Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: michael.breakspear{at}newcastle.edu.au Abstract Full Text Info/History Metrics Supplementary material Data/Code Preview PDF Abstract Functional interactions between cortex and hippocampus play a central role in cognition and are disrupted in major neurological disorders, but the mechanisms underlying coordinated cortico-hippocampal dynamics are poorly understood. We address this challenge using neural field theory, a biophysically-grounded framework for modelling large-scale neural dynamics. We first show how the autonomous activity of cortex and hippocampus emerge from corticothalamic and hippocampo-septal feedback loops, respectively, giving rise to cortical alpha and hippocampal theta rhythms. We next integrate these two systems through topologically and topographically informed coupling between cortex and hippocampus. Weak coupling yields spatially precise correlations between cortical and hippocampal activity, consistent with neurophysiological recordings. Stronger coupling pushes both the cortex and the hippocampus toward criticality, triggering state transitions and mode mixing, such that activity propagates across spatial scales and reorganizes both cortical and hippocampal dynamics. These disruptive, unstable processes also provide an explanation for the frequent involvement of the hippocampus in seizures. This prediction is validated using intracranial electroencephalographic data from human patients with focal onset epilepsy. Together, these results establish a geometrically and biophysically grounded framework that gives a unifying account of large-scale cortico-hippocampal dynamics and provides a physically principled foundation for studying other distributed brain systems. I. INTRODUCTION The hippocampus is pivotal to many cognitive functions [ 1 ], including memory formation and retrieval [2– 8], spatial navigation [ 9 – 13 ], and contextual processing [ 14 ]. These functions do not emerge from hippocampal activity in isolation. Instead, they arise from dynamic, topographically organized interactions between the hippocampus and the neocortex [ 7 , 15 ]. These cortico-hippocampal interactions integrate information across distributed neural systems and facilitate the encoding, consolidation, and retrieval of memories [ 16 ]. Despite significant advances in understanding the individual contributions of the hippocampus and the neocortex, the mechanisms governing their interactions remain unclear [ 4 , 7 , 8 ]. Addressing this gap requires a clear understanding of the anatomical pathways through which corticohippocampal interactions occur. The hippocampus has widespread, reciprocal connections with neocortical regions, with many connections passing via the entorhinal cortex [ 17 – 22 ]. These connections are topographically organized, with specific cortical areas projecting to distinct regions of the hippocampus and vice versa [ 21 , 23 – 25 ]. This bidirectional flow of information creates a dynamic loop, enabling the hippocampus to act as a hub for coordinating and linking distributed cortical activity [ 22 , 26 ]. Thus, the relationship between the hippocampus and cortex is characterized by a spatially organized, functionally adaptive, and reciprocally interconnected architecture that underpins their shared role in cognition [ 27 ]. The large-scale interactions between the cortex and the hippocampus are also shaped by the intrinsic rhythms of each structure, most prominently cortical alpha and hippocampal theta. Cortical alpha is sustained by recurrent corticothalamic loops, where the thalamus acts as a central relay for feedback and modulation of cortical activity [ 28 – 34 ]. Hippocampal theta activity arises through interactions with the nearby medial septum, which provides rhythmic excitation and inhibition that shapes the theta activity [ 35 , 36 ]. Together, these feedback loops of corticothalamic and hippocampo-septal circuits provide the anatomical substrate for the intrinsic cortical and hippocampal dynamics. A topologically and topographically informed cortico-hippocampal coupling is then required to integrate these two systems and explore their emergent, interconnected dynamics. While these anatomical pathways form the structural basis for cortico-hippocampal interactions, they cannot fully account for the complex, dynamic properties of both the cortex and hippocampus. The brain operates as a dynamic and highly interconnected system [ 37 , 38 ], where interactions between neocortical, archicortical, and subcortical structures give rise to complex spatiotemporal dynamics underlying both healthy cognition [ 39 – 46 ] and pathological brain states [ 47 – 49 ]. Capturing the large-scale dynamics in such systems requires a biophysical modeling framework that scales beyond traditional microscopic, spiking neural models. Prevailing macroscopic “mean-field” approaches rely either on neural mass models (NMMs) or neural field theory (NFT) [ 37 ]. NMMs approximate brain regions as discrete nodes connected via empirical structural networks [ 50 – 54 ] and have been used extensively to study cortical and subcortical interactions, including wave-like activity [ 55 , 56 ]. NMMs adopt a node-based formulation, with connections between cortical regions incorporated as discrete node-to-node edges, informed by structural connectivity [ 38 ]. However, NMMs fail to capture the geometric properties of large-scale neural structures such as the folds and undulations of cortex and the curved “seahorse” shape of the hippocampus. Recent work has highlighted the key constraining role of this physical geometry on neural activity, arising in part from horizontal interactions within the curved cortical mantle that are not reliant on long-range myelinated connections [ 57 ]. NFT explicitly models population-level activity as it unfolds over these continuous spatial geometries, embedding surface curvature and neighborhood preservation directly into its equations [ 58 – 63 ]. Hence, this geometrically grounded formulation allows spatially structured phenomena to emerge naturally from the dynamics, with physiologically-derived parameters incorporating key neurophysiological properties [ 63 – 67 ]. In NFT, a continuous distribution of neural activity is treated as a ‘field’ that propagates across the folded cortical surface, providing valuable insights into the underlying biophysical processes that generate large-scale dynamics [ 62 , 63 , 67 ]. Prevailing formulations of NFT have primarily focused on modeling the corticothalamic system, where reciprocal loops between the cortex and thalamus explain the emergence of neocortical rhythms. [ 62 , 63 , 67 – 69 ]. Given accumulating evidence for waves of field-like activity in the hippocampus [ 70 , 71 ], we extend the NFT framework to model hippocampal activity and its interactions with the medial septum and cortex. Separate neural fields are first defined for the hippocampus and cortex, considered in isolation, capturing each of their distinct functional properties and reciprocal connections with the septum and the thalamus, respectively. The intricately folded structure, smaller size relative to the cortex, and distinctive topological organization of the hippocampus complicates the mapping of activity between cortical and hippocampal surfaces within geometrically informed frameworks. To overcome this challenge, we implement a geometrically-constrained coupling between the cortical and hippocampal surfaces using quasi-conformal mapping [ 72 ]. The ensuing dynamics are then characterized by parametrically increasing cortico-hippocampal interactions, and comparing them to healthy and pathological neural recordings. Modeling cortico-hippocampal interactions with NFT is hence shown to provide a principled framework for understanding their shared role in cognition and generating testable hypotheses about their disruption in neurological and psychiatric disorders. II. RESULTS A. Overview We first describe and numerically simulate NFT for the corticothalamic and hippocampo-septal systems, each considered independently. Cortical and hippocampal geometries are modeled as continuous, folded surfaces embedded in anatomical space. The thalamus and septum are incorporated via distributed feedback circuits that capture their reciprocal connections with the cortex and the hippocampus, respectively. Neural fields are defined by coupled integro-differential equations that describe population-level excitatory activity. Numerical integration is then performed in the geometric space of each (cortical and hippocampal) surface using modal decomposition with geometric eigenmodes. This approach permits a low-dimensional representation of the neural fields, enabling computationally efficient simulations. These simulations, performed independently for each structure, reproduce key spectral features of nonpathological cortical alpha and hippocampal theta rhythms. To investigate how interactions between the cortex and the hippocampus influence their dynamics, we introduced spatially structured coupling between the cortical and hippocampal surfaces. Quasi-conformal mapping [ 72 ] is employed to project the distinct cortical and hippocampal geometries into a common two-dimensional domain. This transformation preserves local topological features and facilitates a spatially structured and proximity-based coupling. By parametrically increasing the strength of these inter-surface projections, the effects of cortico-hippocampal coupling on the emergent dynamics were explored. Increasing the coupling strength is shown to modulate the emergent rhythms on both cortex and hippocampus and yields topologically precise correlations between cortex and hippocampus. Stronger reciprocal coupling initiates a transition to high amplitude, complex, seizure-like oscillations, which are benchmarked against intracranial electroencephalo-graphic (iEEG) recordings from the hippocampus and cortex of human patients with epilepsy. B. Neural Field Theory of Corticothalamic Activity NFT describes the evolution of population-level neural activity as continuous fields defined over the geometry of a surface. In the corticothalamic system, NFT represents the cortical activity as a field sustained by external input and thalamic feedback. The evolution of this field on the cortical surface is governed by a damped wave equation [ 57 , 61 , 73 ] as follows, where c denotes the variables and parameters associated with the corticothalamic system and e denotes the excitatory population. The spatial operator ∇ 2 captures the propagation of activity across the folded cortical surface, with temporal damping rate γ c . The parameter r c denotes the characteristic range of cortical axonal projections, assumed to follow an isotropic exponential distance rule (i.e., probability and weight of connection between two locations decrease as an exponential function of their distance). This damped wave equation models the propagation of the cortical excitatory axonal field at location r and time t . This field is sustained by the source term , representing the mean firing rate of excitatory cortical neurons. In total, the corticothalamic system consists of four neural populations: local excitatory ( e ) and inhibitory ( i ) cortical populations, along with the reticular ( r ) and relay nuclei ( s ) of the thalamus ( Fig. 1a ). For these four neural populations, the mean firing rate is derived from the local mean transmembrane potential through a sigmoidal activation function, where represents the maximum firing rate of neural population j ∈ { e, i, r, s }. The parameters and represent the mean firing threshold and its standard deviation, respectively. The potential arises from the sum of all the presynaptic incoming activity , filtered through synaptic and dendritic processes. The governing equations for can be written as, where α c and β c represent the synaptodendritic time constants. Finally, the neural field equations are completed by incorporating the conversion of outgoing field potentials into the incoming presynaptic inputs after propagating across the cortex or between the cortex and the thalamus. The sum of all incoming activity for each population is given by, where measures the synaptic connectivity strength to population a from population b , as indicated by the subscripts. Consistent with classical NFT [ 61 , 62 , 73 ], is the inhibitory cortical field and locally follows the excitatory cortical field without generating spatially propagating fields. The expression for the inhibitory pop-ulation is analogous to and is omitted for brevity. These interactions complete a closed set of equations, whereby the field term on the cortical surface influences the transmembrane potential , which in turn modulates the source term feeding back into the neural field equation. Note also that the corticothalamic equations comprise a feedback loop between cortex and thalamus, with a time-delay ( τ c ) between these structures. ϕ n 0 represents a baseline nonspecific input and η c is a stochastic input term. Using this framework, corticothalamic dynamics were simulated using previously published physiologically informed parameters ( Table I , Methods section), capturing realistic wave speeds, damping rates, firing rates, neural gains, and synaptic time constants [ 62 , 63 , 67 , 73 ]. Numerical integration of the Equation 1 was performed via modal decomposition, whereby cortical activity at each integration point is projected onto geometric eigenmodes of the cortex ( Fig. 1b and Methods section A). This modal decomposition enables cortical activity to be expressed as a sum of spatial eigenmodes with time-varying amplitudes, providing a compact spatiotemporal description of system dynamics. In doing so, it naturally incorporates the influential role of cortical geometry on large-scale neuronal dynamics [ 57 , 69 , 74 ]. Lower-order eigenmodes correspond to large-scale patterns of activity, whereas higher-order modes capture finer wavelength patterns ( Fig. 1b ). View this table: View inline View popup Download powerpoint TABLE 1. Cortical model parameters. Download figure Open in new tab FIG. 1. Corticothalamic system dynamics. a , Schematic of the corticothalamic model where the letters denote the interactions between the excitatory ( e ), inhibitory ( i ) populations of the cortex and the reticular ( r ) and relay ( s ) nuclei of the thalamus. b , Cortical activity is decomposed into geometric eigenmodes, ensuring a compact spatio-temporal representation. c , Representative eigenmode amplitude time series of the 2nd and 110th eigenmodes illustrate the evolution of cortical activity at different spatial scales. d, e , The average power spectral density (PSD) across all eigenmodes amplitude time series ( d ) and the cortical vertices time series ( e ) is shown in dark blue. The light blue shading indicates ± 1 standard deviation on the logarithmic y-axis. Lower error bounds appear stretched toward zero because small absolute variations near low power values are exaggerated by the logarithm. f , Frequency-specific eigenmode contributions, visualized as a heatmap, highlight the spectral signatures of each eigenmode. g , Dynamic synchrony measure (DSM), computed as the windowed maximum lagged cross-correlation between eigenmode time series, reflects weak correlations across spatial scales. Having defined the model, we now explore its dynamic behavior. The temporal evolution of two representative eigenmodes ( Fig. 1c ) illustrates how the model captures the spatiotemporal dynamics shaped by cortical geometry. These resulting cortical dynamics exhibit an alpha-band (≈ 10 Hz) peak in the power spectral density (PSD) of both vertex-level ( Fig. 1d ) and eigenmode time series ( Fig. 1e and 1f ). Alpha rhythms in this frequency range are a robust feature of resting-state cortical activity, widely documented in human EEG and MEG studies [ 51 , 64 ]. Windowed maximum lagged correlations across all cortical eigenmodes ( Fig. 1g ) reveal that longer wavelength and finer wavelength patterns evolve with minimal mutual influence, indicating localized organization and weak global synchrony. Localized spatial correlations arising from relatively weak global synchronization are hallmarks of nonpathological cortical dynamics [ 75 , 76 ]. Hence, the eigenmode-based simulation retains the key spectral features of cortical activity [ 62 , 63 ], while providing an efficient representation of its spatiotemporal dynamics with the activity of over 29,000 vertices effectively encoded by just 110 cortical eigenmodes. C. Neural Field Theory of Hippocampal Activity The accumulating evidence of large-scale spatiotemporal patterns in the hippocampus [ 77 – 79 ] along with intrinsic geometric organization of the hippocampus [ 4 , 9 ] makes NFT a viable framework for simulating hippocampal dynamics. Hence, we extend the NFT framework to the hippocampo-septal system to verify its ability to simulate hippocampus-specific activity, , and spectral signatures. In particular, attention was given to hippocampal theta rhythms, which are spatiotemporally organized and strongly modulated by the septum [ 35 , 36 , 77 , 80 ]. Hippocampal geometry is embedded into this framework using a population-averaged surface mesh [ 81 ], from which geometric eigenmodes are derived. The first 25 of these eigenmodes are used for modal decomposition during our simulations, capturing large-scale spatial patterns down to ≈2–3 mm resolution. To maintain notational consistency and avoid introducing unnecessary variables, the NFT equations for the hippocamposeptal system adopt the same mathematical structure as those used for the corticothalamic system. Variables and parameters associated with the hippocampo-septal system are distinguished using h instead of c , indicating their correspondence to hippocampal or septal populations. The evolution of the hippocampal excitatory axonal field is governed by a damped wave equation similar in structure to the wave equation in the corticothalamic system ( Equation 1 ), where the parameters γ h and r h represent the temporal damping rate and characteristic spatial range of hippocampal excitatory neural populations. Comparable to the corticothalamic system, the hippocampus and medial septum also form a feedback loop that plays a pivotal role in shaping the spatiotemporal dynamics of the hippocampal theta rhythm [ 1 , 35 , 82 – 84 ]. The time delay in this loop is represented by τ h ( Table II , Methods section). The remaining equations governing source terms and transmembrane potentials follow the same mathematical structure as the corticothalamic system [Equations (2), (3), and (4)], with parameters adjusted for the hippocampo-septal system. The subscripts e and i in the hippocampo-septal system denote excitatory and inhibitory populations within the hippocampus, while r and s refer to inhibitory and excitatory populations in the medial septum ( Fig. 2a ). Throughout the manuscript, hippocampal dynamics are shown in red and cortical dynamics in blue across all spectral and correlation visualizations. Download figure Open in new tab FIG. 2. Hippocampo-septal system dynamics. a , Schematic of the hippocampo-septal model showing reciprocal connections between the hippocampus and the septum. b , Hippocampal activity is decomposed into eigenmodes, providing a compact representation of its spatiotemporal dynamics. c , Time series of the 2nd and 25th eigenmodes illustrate the evolution of hippocampal activity at different spatial scales. d, e , Average power spectral density (PSD) estimates for all the hippocampal vertices in d and all the eigenmode amplitude time series in e reveal theta-band activity (3-8 HZ). Dark lines indicate the mean PSD, and shaded regions show ± standard deviation across vertices ( d ) and modes ( e ). f , Frequency-specific eigenmode contributions highlight their individual spectral signatures. g , Dynamic synchrony measure, quantified using a windowed maximum lagged cross-correlation between non-zero eigenmode time series, indicates weak correlations across hippocampal eigenmodes. The rationales for parameter selection and scaling in the hippocampo-septal NFT are described in the Methods section. View this table: View inline View popup Download powerpoint TABLE 2. Hippocampal model parameters. Simulating this hippocampo-septal system within the NFT formalism yields robust theta-band activity (≈3–8 Hz). This is evident in both the representative hippocampal eigenmode amplitude time series ( Fig. 2c ) and the PSD of the surface vertex time series ( Fig. 2d ) and the eigenmode time series ( Fig. 2e and 2f ). Notably, the absence of strong correlations between different hippocampal eigenmodes ( Fig. 2g ) implies limited interactions across modes corresponding to different spatial scales during non-pathological conditions. Alpha-band activity in the cortex and theta-band activity in the hippocampus have been widely reported in empirical observations [ 4 , 35 ]. Hence, these simulations demonstrate that the NFT framework, when applied to anatomically grounded corticothalamic and hippocamposeptal systems, yields physiologically realistic rhythms. D. Cortico-Hippocampal Coupling The corticothalamic and hippocampo-septal systems are next combined into a single integrated framework ( Fig. 3a ) by introducing a bidirectional coupling between the cortical and hippocampal surfaces (denoted with C ch and C hc in Fig. 3a ). Since neural fields evolve on continuous two-dimensional cortical and hippocampal manifolds, coupling these systems requires a shared representation of both surfaces to enable spatially structured interactions. Download figure Open in new tab FIG. 3. Coupled cortico-hippocampal system. a , The bidirectional connectivity between the cortex and hippocampus is represented by their coupling constants C ch and C hc . b , Visualization of the hippocampus → cortex coupling enabling smooth field transfer between surfaces via quasi-conformal mapping. c – f , Detailed illustration of the mechanism underlying hippocampus → cortex field propagation. Both cortical and hippocampal surfaces are mapped onto a common 2-D domain via a quasiconformal transformation ( d and e , respectively). The target cortical vertex ( f ) and its location on the 2-D cortical map ( e ). The corresponding hippocampal region ( d ) projecting to the target cortical vertex is computed via the coupling function, which minimizes distance in the 2-D domain between the target cortical vertex and the hippocampal region. The color gradient in this region represents the coupling strength, peaking at the center (red) and exponentially decaying outward. g – j , Illustration of the field propagation from the hippocampal surface to the cortical surface performed with the proposed coupling scheme. The same methodology can be implemented in the opposite direction (cortex to hippocampus) by swapping source and target sheets. To address this, we applied a quasi-conformal transformation to map both the cortical and hippocampal surfaces onto a common two-dimensional Euclidean domain ( Fig. 3b, 3d , and 3e ). This projection to a shared coordinate system enables a biologically inspired, topographically constrained coupling, where nearby regions on one surface preferentially interact with nearby regions on the other, aligned along anterior–posterior gradients [ 24 ]. To implement smooth, spatially localized interactions across the shared coordinate domain, a coupling kernel is introduced that enables field propagation between the hippocampal and cortical surfaces. Each cortical vertex receives input from a localized region of the hippocampal surface ( Fig. 3b ), determined by an exponentially decaying coupling kernel. Similarly, hippocampal vertices receive input from cortical regions through analogous kernels. Specifically, for each target vertex on the cortical surface, the coupling kernel is centered on the nearest vertex on the hippocampal surface in the shared 2-D Euclidean domain( Fig. 3b ). Neural field values within this kernel are exponentially weighted based on distance and then projected onto the target cortical location. As the cortical target moves, the kernel smoothly traverses the hippocampal surface, preserving spatial continuity. This establishes a continuous, bidirectional mapping where neighboring vertices on one surface project to neighboring vertices on the other. Representative coupling kernels on the hippocampal surface are illustrated in Fig. 3b, 3c , and 3d . Warmer colors on these kernels indicate stronger coupling weights, and the combined projection of cortical activity from these kernels onto their corresponding cortical vertex is illustrated in Fig.3b . The blue dots in Fig.3e and 3f represent the cortical vertex onto which the activity from the kernel in Fig.3c or 3d is projected. Implemented bidirectionally, this coupling scheme preserves local neighborhood structure on both surfaces, avoiding discontinuities in field propagation across surfaces ( Fig. 3g–j ). Implemented this way, the quasi-conformal maps enforce biologically inspired anterior–posterior alignment, whereby anterior cortical areas preferentially couple to anterior hippocampal regions and posterior cortical areas to posterior hippocampal regions (see Supplementary Materials Fig. S1–S3). To formalize this coupling between the cortical and hippocampal fields, the uncoupled neural field equations are extended by incorporating spatially structured inter-surface interaction terms W ch ( r, r ′ , t, t ′ ) and W hc ( r, r ′ , t, t ′ ). The wave equations for cortical and hippocampal activity introduced earlier can be compactly written as follows, where D c and D h are the cortical and hippocampal differential operators introduced in Equation 1 and 5, respectively. The coupled system dynamics are then obtained by adding the coupling term in the source terms of Equations 1 and 5, yielding the full evolution equations, where the coupling functions W ch and W hc are constructed using the quasi-conformal mapping, projecting both the cortical and hippocampal surfaces onto a common two-dimensional Euclidean space ( Fig. 3b , Fig. 3d and 3e ). Spatial coordinates r and r ′ denote locations on the target and the source surface, respectively. Given the finite conduction velocities along axonal projections, neural activity originating at a vertex on one surface (source surface) reaches the other surface (target surface) after a characteristic delay ( τ ch ). While such delays can in principle depend on distance between vertices, we assume a uniform, constant delay τ ch for tractability. Under the assumption that this delay is uniform and constant for cortico-hippocampal interactions, the temporal coupling function naturally simplifies to a Dirac delta function, δ ( t − t ′ − τ ch ). Thus, the full spatiotemporal coupling function in Equations 8 and 9 can be expressed as the product of spatial ( w ch and w hc ) and temporal coupling terms, scaled by the coupling strengths C ch and C hc . The cortico-hippocampal interactions are predominantly excitatory in nature [ 7 , 15 , 21 ]. Hence, C ch and C hc are positive constants representing the strength of excitatory projections from hippocampus to cortex and vice versa. The coupling terms can now be decomposed as follows, To ensure spatially localized interactions that decay smoothly without introducing discontinuities at the kernel boundaries, we adopt a radially symmetric, exponentially decaying coupling kernel of the form . This choice naturally prioritizes proximal interactions in the shared Euclidean domain while maintaining mathematical tractability. κ is the parameter governing how quickly the interaction decays with distance in the shared 2D coordinate space. By applying the identity, , the time integral in the coupling equation simplifies to, This completes the set of equations for the coupled cortico-hippocampal system. In this framework, the field from the source surface is transported to the target surface via a spatial convolution with the coupling kernel . Having achieved an integrated cortico-hippocampal system, we first studied the impact of slowly increasing the inter-surface coupling strength ( C ch and C hc ) on the unified cortico-hippocampal dynamics. Theoretical analysis (Supplementary Materials S2) suggests that bidirectional cortico-hippocampal coupling enhances crosssurface mode–mode interactions and leads to systematic shifts in dominant system frequency. Numerical integration shows that increasing coupling between the two surfaces initially sharpens and shifts the alpha and theta spectral peaks toward higher frequencies ( Fig 4a ). These are characteristics of systems nearing critical transitions unless countervailing changes are made [ 85 ]. To further explore these trends, the strength of corticohippocampal coupling was increased in steps of 0.001 while keeping all other parameters for the corticothalamic and hippocampo-septal systems fixed within their respective alpha and theta rhythm regimes ( Tables I , II ). Visualizing the spectral content of the resulting dynamics as a heat map shows that stronger cortico-hippocampal coupling enhances synchrony and shifts activity toward faster frequencies on both surfaces ( Fig. 4b ). However, for excessive coupling between the two surfaces ( C ch , C hc > ≈ 0.3), these peaks broaden then disperse. Download figure Open in new tab FIG. 4. Effects of cortico-hippocampal coupling on spectral dynamics and spatial correlations. a , Power spectral density (PSD) of cortical (blue) and hippocampal (red) eigenmode time series at three coupling strengths ( C ch = C hc = 0 ( a -(i) ), 0.01 ( a -(ii) ), 0.015 ( a -(iii) )), showing progressive sharpening and upward shifts in dominant frequencies (arrows). b , PSD heatmaps illustrating the evolution of peak oscillation frequencies in the cortex (top, blue heatmap) and hippocampus (bottom, red heatmap) as a function of bidirectional coupling strength. c , Spatial specificity of cortico-hippocampal interactions, quantified by cross-correlation coefficients (CCC) between time series from a representative cortical region (blue patch) and all hippocampal vertices. In the uncoupled state ( C ch = C hc = 0), correlations are weak and spatially diffuse, whereas stronger coupling (ii, iii) induces locally organized synchrony between topographically aligned cortical and hippocampal regions. d , Similar patterns of topographically organized cross-structure correlations emerge when analyses are extended to multiple cortical regions. We next validated whether the structural mapping between the cortex and hippocampus yields a corresponding topographically specific functional mapping. Specifically, we checked whether the simulated cortical activity in the coupled cortico-hippocampal system aligns preferentially with topographically linked hippocampal regions and vice versa. To quantify the spatial specificity of cortico-hippocampal coupling, maximum lagged cross-correlations were calculated between the time series from hippocampal and cortical surfaces (see Methods). In the absence of cortico-hippocampal coupling, correlations between a single representative cortical region and hippocampal vertices remain low and spatially unstructured ( Fig. 4c (i) and 4d (i) ). With increasing corticohippocampal coupling ( C ch = 0.01 and C ch = 0.015), this cortical region exhibits localized synchronization with specific hippocampal areas ( Fig. 4c (ii) and 4c (iii); 4d (ii) and 4d (iii) ). Notably, cortical and hippocampal regions occupying similar locations within their respective mapped domains display enhanced correlations. This pattern also reflects the anterior-posterior organization embedded in our coupling scheme. Extending this analysis to two representative cortical regions reveals that each synchronizes with a distinct, localized region of the hippocampal surface ( Fig. 4d ). These results show that cortico-hippocampal coupling promotes topographically organized synchrony while preserving local neighborhood structure, rather than inducing a global increase in coherence. Taken together, the upward frequency drift, peak sharpening, and increased inter-system coherence are consistent with the approach to a critical transition. These changes, observed with increased coupling strength, suggest a potential mechanism by which the cortico-hippocampal system may shift from healthy dynamics into pathological states [ 86 ]. In the following section, this hypothesis is tested by systematically varying the parameters that control excitability within the model. The resulting dynamics are then compared to empirical iEEG data acquired from human patients during epileptic seizures. E. Cortico-Hippocampal Seizure Modeling Recurrent loops in large-scale neural architectures, such as the cortico-hippocampal system, are known to enhance mutual gain through positive feedback, thereby increasing the system’s susceptibility to dynamical instabilities [ 63 , 87 ]. These instabilities have been associated with transitions to seizure-like states in both computational models and empirical studies [ 88 ]. At the cellular level, glutamatergic neurons play a pivotal role in initiating and propagating epileptic seizures [ 89 ]. In line with this, glutamatergic transmission in the hippocampus has been implicated in driving hyperexcitability, with elevated extracellular glutamate levels observed in epileptogenic regions [ 90 , 91 ]. During seizure-like transitions, increased functional coupling and phase synchrony between cortical and hippocampal networks have also been observed [ 92 ]. In our NFT model, the parameter modulates the level of excitation in the hippocamposeptal feedback loop. The strength of excitatory corticohippocampal coupling is controlled by C ch and C hc . Together, these parameters define the excitatory feedback landscape across both local and long-range hippocampal loops and can be dynamically varied to model a representative hippocampal seizure consistent with focal onset Mesial Temporal Lobe Epilepsy (MTLE). Another ubiquitous feature of many seizures [ 47 , 93 ] is their dynamically changing spectral content. Individual hippocampal seizures exhibit diverse spectral dynamics, but a recurring feature is the presence of spectral chirps progressive shifts in dominant frequency - as the seizure progresses. To assess whether our cortico-hippocampal model can reproduce these features, we simulate a representative hippocampal onset seizure ( Fig. 5 ), consistent with MTLE, by increasing excitability in all loops involving the hippocampus, i.e., the hippocampo-septal feedback loop and the bidirectional cortico-hippocampal connections ( C ch and C hc ). The excitability in the corticothalamic loop was held constant, although it can be modulated in order to capture seizures involving broader cortical recruitment. An increase in the amplitude of hippocampal and cortical activity can be observed after the simulated seizure onset ( Fig. 5a and 5d ). This amplitude increase is also accompanied by upand down-chirps in frequency ( Fig. 5b and 5e ) captured by the hippocampal and cortical spectrograms as the seizure evolves. These results suggest that the transition to a MTLE seizure could reflect a critical transition in the cortico-hippocampal system, where a small change in network parameters yields large-scale dynamical reconfigurations. In particular, we find that recurrent excitatory interactions involving the hippocampus form a minimal dynamical substrate capable of generating and sustaining seizure-like activity. Download figure Open in new tab FIG. 5. Simulated hippocampal-onset seizure consistent with Mesial Temporal Lobe Epilepsy (MTLE). a, d Representative time series from hippocampal and cortical surfaces. Average spectrograms for the hippocampal ( b ) and cortical ( e ) surfaces showing frequency modulation as the system parameters evolve. Mode-mode coupling between the hippocampal ( c ) and cortical ( f ) eigenmodes during a pre-seizure (panel - (i)) 30-second epoch (270 s < t < 300 s) and during peak hippocampal excitability and cortico-hippocampal coupling (panel - (ii)) (316 s < t < 345 s) window. g , Evolution of the hippocampal excitability and the cortico-hippocampal coupling ( C ch and C hc ) during the pre-seizure and seizure domains. h , Mode-mode correlations between the cortical and hippocampal eigenmodes (across surface mode dynamics) during the pre-seizure epoch (panel - (i)) and during seizure activity (panel - (ii)). We next examined how these transitions reorganize multiscale activity across the two structures. To achieve this, we studied correlations between eigenmode time series within and between cortex and hippocampus, which provides an insight into the spatial restructuring that accompanies seizure onset. Prior to seizure onset, these multiscale correlations are relatively weak and uniformly distributed across eigenmodes within and between the cortex and hippocampus ( Fig. 5c (i), 5f (i) , and 5h (i) ). These indicate a state of relatively desynchronized, non-pathological activity. During the seizure epoch, a marked increase in these correlations occurs. Within the hippocampus, synchrony strengthens across multiple eigenmodes ( Fig. 5c (ii) ). A similar increase occurs among cortical eigenmodes, but this effect is predominantly confined to low order modes ( Fig. 5f (ii) ). Additionally, cross-structure coupling intensifies, with cortical low-order modes showing enhanced temporal alignment with hippocampal modes ( Fig. 5h (ii) ). This simulated seizure provides an in silico demon-stration of how cortico-hippocampal dynamics shift into pathological regimes. However, empirical seizures in individual patients exhibit substantial dynamic and spectral heterogeneity [ 47 ]. We next tested the model’s ability to reproduce key spectral dynamics of individual empirical seizures ( Fig. 6 ). Patient-inferred parameter trajectories from empirical iEEG data were used to inform the model. The data used for this purpose consist of iEEG recordings from three patients with hippocampal onset seizures, analysed following Institutional Human Research Ethics approval (see Methods and Supplementary Material S1.3). For each patient, the seizure onset and termination times were annotated by an epileptologist. All patients had iEEG electrode contacts in the hippocampus and cortex based on clinical considerations, with approximately equal numbers in each region. However, because the hippocampus is smaller, this results in a more comprehensive coverage of the hippocampus. Download figure Open in new tab FIG. 6. Empirical and simulated dynamics of hippocampal focal seizures (Patient id: P2). a , Intracranial EEG (iEEG) recordings from three hippocampal electrodes and c , four cortical electrodes in a patient with hippocampal epilepsy. Dotted red lines mark the onset and termination of the seizure as annotated by an epileptologist. b, d , Average short-time Fourier transform (STFT) spectrogram of the hippocampal ( b ) electrodes and ( d ) cortical electrodes from the cingulate cortex. e , Evolution of synchrony across unique hippocampal electrode pairs (pink), cortical electrode pairs (blue), and cortico–hippocampal electrode pairs (black). f , System parameter trajectories inferred from the dynamic synchrony evolution in ( e ): hippocampal and cortical excitatory drive ( and ) and cortico-hippocampal coupling ( C ch and C hc ). g, i , Simulated time series from representative hippocampal ( g ) and cortical ( i ) vertices. h, j , Simulated spectrogram from a representative hippocampal ( h ) and cortical vertex ( j ). Cortical coverage varied across patients, encompassing regions such as the cingulate cortex, fusiform area, temporal pole, and orbitofrontal cortex. iEEG recordings were sampled at 500 Hz, providing a detailed view of the temporal evolution of frequency content and synchronization dynamics across hippocampal and cortical regions. (Demographics and clinical details for all three patients are provided in Supplementary Table S1). In a representative patient’s seizure dynamics an increase in iEEG amplitude along with dynamic frequency content occurs during the seizure epoch across the hippocampus ( Fig. 6a ) and the cortex ( Fig. 6c ). An upchirp is observed in the hippocampus at seizure onset ( Fig. 6b ), followed by a gradual frequency reduction toward seizure termination in both the cortex ( Fig. 6d ) and the hippocampus ( Fig. 6b ). A similar pattern was observed in other patients, highlighting the non-stationary frequency dynamics that characterize MTLE seizure activity (Supplementary Figures S4 and S5). To quantify seizure-related changes in synchronization dynamics, a dynamic phase synchrony measure (DSM) between electrode pairs was computed using a windowed, lagged cross-correlation approach. DSM was estimated independently for three anatomical groupings: (i) hippocampus–hippocampus (H–H), (ii) cortex–cortex (C–C), and (iii) cortex–hippocampus (C–H) electrode pairs. DSM time series from these three anatomical groupings show synchronization evolution across all the electrode pairs during seizures ( Fig. 6e ). In this representative patient, hippocampal synchrony increased sharply around the seizure onset ( Fig. 6e , pink trajectory), suggesting a pathological coordination of neural activity within the hippocampus. This increase in synchronization was accompanied by a progressive rise in corticohippocampal coupling ( Fig. 6e , black). Both hippocampal and cortico-hippocampal synchrony then decreased toward seizure termination, reflecting their transient and dynamic involvement in seizure propagation. In contrast, cortico-cortical correlations (iEEG electrodes in the cingulate cortex) remained relatively constant following a brief increase after seizure onset ( Fig. 6e , blue). These observations suggest that the seizure dynamics in this patient were dominated by hippocampal and corticohippocampal synchronization, with minimal involvement of cortico-cortical interactions. Temporal trajectories of DSM from each anatomical group were used to inform the temporal evolution of model parameters governing excitability and coupling in the NFT. In the purely in silico simulation ( Fig. 6 ), only , and C hc were modulated, to capture a canonical hippocampal-onset seizure. However, the brief rise in cortico-cortical synchrony observed here motivated its inclusion. Our model accommodates such adjustments, enabling patient-specific dynamics to guide parameter evolution. Hence, , and C hc evolution was guided by empirical DSM values to reproduce the dy-namics observed in the patient’s data. The evolution of each parameter was defined using simple, piecewise linear functions whose timing and amplitude were guided by the empirical synchrony trajectories for each anatomical grouping, allowing a subject-specific seizure simulation ( Fig. 6f ). The simulated seizure dynamics employing these parameter trajectories reproduce core spectral features observed in the empirical seizure, i.e., the up- and down-chirps in frequency ( Fig. 6g–j ). Together, these empirically constrained simulations show that time-varying changes in hippocampal excitability and cortico-hippocampal coupling drive hallmark seizure features such as spectral chirps and transient synchrony within and across structures. To further validate this approach, data from two additional patients with hippocampal epilepsy was extracted and their respective synchronization profiles were used to drive parameter evolution. Simulated dynamics for both patients captured key spectral patterns observed in their recorded seizures (Supplementary Figures S4 and S5). III. DISCUSSION Coordinated activity in the cortex and hippocampus supports cognitive functions and underlies pathological brain states, but a grounded theory of corticohippocampal interactions has been lacking. To address this, we extend the formalism of neural field theory (NFT) to construct a physically grounded model of cortico-hippocampal dynamics by embedding both structures within their native geometries and linking them through topologically and topographically constrained interactions. NFT naturally models the evolution of population-level brain activity as a continuous field across the cortical and hippocampal geometries, capturing both local dynamics and global constraints [ 58 , 73 , 87 ]. A bidirectional coupling between these two structures was introduced by co-registering them in the same space via quasi-conformal mapping. This approach preserves the topological continuity and achieves topographical alignment across the anterior-posterior gradients. As a result, spatially localized perturbations in one structure propagate along anatomically plausible pathways to the other, providing a unifying framework for modeling both healthy coordination and seizure dynamics. By tuning physiologically interpretable parameters within this framework, we observe a variety of system behaviours that align with empirical observations. Weak reciprocal coupling between cortex and hippocampus leads to a sharpening of intrinsic rhythms—alpha in the cortex, theta in the hippocampus—suggesting an enhancement of spectral precision through modest inter-areal synchrony. As the coupling strength is further increased, the system approaches a critical regime marked by amplification and increase of dominant frequencies, culminating in spectral chirps reminiscent of those observed during epileptic seizures [ 47 , 93 ]. These emergent transitions reflect a reconfiguration of the underlying dynamical landscape, wherein cortico-hippocampal interactions reshape local instabilities and facilitate cross-scale entrainment, conceptually consistent with recent analyses of whole field recordings [ 75 ]. In line with the theoretical predictions presented in the supplementary materials, the model demonstrates how modest changes in coupling can tip the system from healthy coordination to pathological resonance. The present work adds to the emerging body of work using large-scale biophysical models to understand the onset and dynamics of epileptic seizures [ 62 , 94 – 96 ]. Our work aligns with recent work that finesses such models to the specific seizure waveforms seen in individual patients [ 97 ] and tracks their time course through parameter space [ 98 – 100 ]. Notably, our work brings a specific focus on cortico-hippocampal mean field dynamics, acknowledging the core role of the hippocampus in adaptive cognition and pathological seizure states. The onset of many seizures is heralded by the appearance of spatially localized high-frequency discharges, although these were not evident in the seizures we studied. Instead, seizure initiation was implemented via an increase in the coupling and gain parameters. The increase in gain enhances excitation in the system and simulates the glutamatergic release observed during the seizure onset [ 89 ]. Future extensions that incorporate customized mean-field models of seizure dynamics [ 101 , 102 ] or cellular-level mechanisms of seizure generation [ 97 ] could help unify microscopic pathophysiology with large-scale brain dynamics. Contextualizing our results within the field of largescale brain modeling highlights the distinct explanatory potential of NFT. High-fidelity reconstructions such as Spaun [ 103 ] and the Digital Brain [ 104 ] offer biological detail but are computationally expensive and difficult to invert. In contrast, data-driven foundation networks [ 105 ] occupy the other end of the spectrum, achieving strong predictive accuracy but offering limited insight into underlying mechanisms. NFT occupies a complementary middle ground, combining computational efficiency with the ability to derive closed-form solutions that treat the brain as a physical dynamical system. This, in turn, links parameters such as gain and intersystem coupling to specific neurobiological processes, enabling principled interpretation of macroscopic brain activity and linking geometry and emergent dynamics in a theoretically transparent manner. The present findings also pave the way for further refinements of the NFT framework to incorporate additional anatomical and physiological details. First, the classical NFT assumes an isotropic and uniform synaptic kernel approximated by the exponential distance rule (EDR) for the axonal projections. Incorporating anatomically informed long-range connections would enable insight into the influence of second-order attributes of the connectome on large-scale brain dynamics. Second, integrating subject-specific cortical and hippocampal surfaces reconstructed from magnetic resonance imaging (MRI), together with individualized connectomes, could transform the framework into a personalized digital twin [ 94 ]. Third, the hippocampal theta activity in our cortico-hippocampal model is generated by the hippocampo-septal feedback loop guided by prior work on the involvement of septum in hippocampal theta [ 35 , 106 ]. However, the hippocampus is a complex structure with local recurrent feedback between hippocampal subfields and the dentate gyrus, along with reciprocal connections to other external structures, including the entorhinal cortex and thalamus [ 20 – 22 , 32 ]. Explicitly incorporating these refinements in subsequent research could further augment the biological accuracy of this framework. Finally, a principled method for model parameter evolution during the individual seizure simulations, such as a variational approach, could also replace the current reliance on visual tuning. IV. METHODS Neural Field Theory (NFT) parameters The system parameters used for the corticothalamic and the hippocampo-septal loop are listed in Tables I and II . The parameters for the corticothalamic system were adopted from previously published papers where the corresponding NFT predicted and reproduced key features of cortical dynamics [ 62 , 63 , 67 , 73 ]. Thalamocortical projections are relatively sparse and topographically organized [ 107 , 108 ]. In contrast, the medial septum forms dense and widespread connections with the hippocampus [ 80 , 109 , 110 ]. To reflect these anatomical differences, the synaptic gain parameters in the hippocampalseptal loop were scaled relative to those of the corticothalamic system. Furthermore, because the hippocampus and septum are in close anatomical proximity, with fastacting GABAergic and cholinergic projections mediating hippocampal theta rhythms [ 35 , 36 ], a shorter feedback delay ( τ h ) was employed than the corresponding corticothalamic delay ( τ c ). This parameter set captures the stronger local coupling and reduced conduction delays characteristic of the hippocampo-septal circuitry while preserving the mathematical structure of the NFT. A. Integration Using the method of separation of variables, the neural field can be decomposed into spatial and temporal components, with spatial modes represented by , the eigenfunctions of the Laplace-Beltrami operator. This leads to an eigenvalue problem for the Laplace–Beltrami operator, where Δ denotes the Laplace–Beltrami operator defined on the cortical (a = c) or the hippocampal (a = h) surface. The solutions form a natural basis for representing activity on curved surfaces and correspond to the geometric eigenmodes of the cortical or hippocampal sheet. Because the eigenmodes form a complete and orthogonal basis for any smooth field defined on the cortical or hippocampal surface, the neural field can be efficiently represented as a weighted sum of eigenmodes, where denotes the time-varying amplitude associated with the n th eigenmode, and N a denotes the number of eigenmodes used in the decomposition. Substituting this expression into the NFT wave equation and decomposing the source term as , the temporal evolution of the system in eigenmode space becomes, The field evolving on the uncoupled cortical/hippocampal surfaces is sustained by the net excitatory and inhibitory input received from local cortical/hippocampal populations and the thalamic/septal nuclei. These inputs are embedded in the source term . To efficiently capture the dominant spatial patterns while maintaining computational tractability, the field on the cortical surface was represented using the first 110 geometric eigenmodes. This truncation corresponds to a minimum spatial wavelength of approximately 40 mm, consistent with recent empirical findings [ 57 ]. For the hippocampus, which is smaller in size, the dynamics were modeled using the first 25 eigenmodes, resulting in a spatial field resolution of approximately 3 mm. Following previous work [ 62 , 96 ], the characteristic range for thalamic nuclei and the medial septum is set to zero ( r a = 0), thereby suppressing local propagation. This simplification allows thalamic populations to be modeled as discrete nodes rather than continuous fields, in contrast to the surface-based treatment of the cortex and the hippocampus. This eigenmode-based formulation enables region-specific control over spatial autocorrelation by adjusting the number of modes used in each structure. For example, the choice of 110 cortical modes and 25 hippocampal modes allows different levels of spatial autocorrelation to be imposed on the cortical and hippocampal surfaces, respectively. B. Coupling between the cortex and the hippocampus The cortical and hippocampal surfaces used in this study are population-averaged anatomical templates. To simulate dynamic cortico-hippocampal interactions, a bidirectional coupling scheme was introduced, treating both the cortex and the hippocampus as continuous twodimensional neural sheets. Let r denote spatial locations on the surface of interest (either cortex or hippocampus), and r ′ denote locations on the surface where the field originated. This convention enables a consistent formulation of interactions across the coupled corticohippocampal system. As per Results Section D, the coupled neural field dynamics are governed by, where D c and D h are the wave operators defined for the cortex and hippocampus, respectively. The functions W ch and W hc describe the spatiotemporal coupling from hippocampus to cortex and vice versa. Source terms and represent thalamic/septal input to the corti-cal/hippocampal surface. To retain spatial specificity and analytical tractability, we assume the coupling is uniform and isotropic and thus only depends on spatial and temporal offsets, where w ch (·) and w hc (·) are spatial kernels, and τ ch is the fixed transmission delay. To construct these spatial coupling kernels, we independently map the cortical and hippocampal surfaces to two-dimensional domains using quasi-conformal mapping [ 72 ]. A quasi-conformal map is a homeomorphism from a 2D manifold (surface) embedded in ℝ 3 to a planar domain, preserving local topology and introducing only bounded angle distortion (with Beltrami coefficient µ ( z ) satisfying | µ ( z ) | < 1). For each surface S a ( a ∈ { c, h } for cortex and hippocampus), we obtain a bijective map ensuring a one-to-one correspondence between the curved surface in 3D and its 2D flattened, planar representation. Notably, this procedure does not define a bijection between cortical and hippocampal surfaces themselves, as their geometries and discretizations generally differ. Instead, the flattening simply provides a consistent 2D domain for subsequent kernel construction and enables proximity-based coupling, where interactions between surfaces are determined by distances between vertices in their respective 2-D embeddings. The coupling strength between a cortical vertex c i and a hippocampal vertex h j is then defined using an exponentially decaying function of Euclidean distance, where is the decay constant, and is the rank of hippocampal vertex h j by distance from cortical vertex c i . Each cortical vertex receives weighted input from its 99 nearest hippocampal neighbors in this shared Euclidean space, and vice versa. Under the assumption of approximately uniform inter-vertex spacing, this enforces a consistent projection area across the surfaces. Similarly sized cortical patches influence each hippocampal location, and vice versa. This implementation supports spatially selective field transport between surfaces. This coupled cortico-hippocampal system was integrated using the same eigenmode-based approach as described previously. The cortical and hippocampal surfaces were each decomposed into geometric eigenmodes, and their respective wave equations were solved in this eigenmode space. The coupling was added to the subcortical input term . This combined input was projected into the eigenmode basis and the coupled wave equations were solved in the eigenmode space. Thalamic and septal populations remained point-based and were integrated independently at each location, consistent with the assumption of zero spatial range. This scheme enabled consistent and computationally efficient simulation of the coupled surface–point system. Normalized amplitude spectrum estimation The spectral content of cortical and hippocampal dynamics was quantified using a single-sided power spectral density (PSD). Given a signal x ( t ) sampled at frequency F s , we first compute its discrete Fourier transform X ( f m ) = FFT[ x ( t )]. The PSD is then estimated as where N is the number of time points, and f m denotes the discrete frequency bins obtained from the FFT. Prior to FFT computation, the time series were detrended and multiplied by a Hanning window to reduce spectral leakage. The PSD was then computed using the absolute squared magnitude of the FFT, normalized by the sampling frequency F s and the total number of points N . Finally, the spectrum was truncated to retain only the positive frequencies (single-sided spectrum). Spatial Specificity of Cortico-Hippocampal Coupling To assess the spatial specificity of cortico-hippocampal coupling, cross-correlation coefficients (CCCs) were calculated between time series from cortical regions of interest (ROIs) and individual hippocampal vertices. Because a single cortical vertex carries negligible functional relevance, a small cortical region—comprising a central vertex and its 99 nearest neighbors was used to approximate a localized population ( Fig. 4c and 4d , blue patches). The maximum lagged cross-correlation between each hippocampal vertex and all 100 vertices in the selected cortical region was computed and averaged to yield a single correlation score per hippocampal vertex. Repeating this analysis for every hippocampal vertex produces a spatial map of how strongly each hippocampal location synchronizes with the chosen cortical patch. representative cortical ROI ( Fig. 4c and 4d ), was defined as the 100 nearest cortical vertices to a selected seed point. That is, for a hippocampal vertex h k and a cortical vertex v i , the lagged cross-correlation was calculated over the entire time series, within a lag range of ±1 s: where and are mean values of the respective time series and d is the temporal lag. The maximal absolute cross-correlation was then calculated as: where Δ d = 1 s. For each hippocampal vertex h k , the average CCC across all the vertices in the cortical ROI vertices was calculated as, where C = 100 is the number of cortical vertices in the ROI C . This produced hippocampal surface maps of , reflecting the strength of coupling between each cortical ROI and the entire hippocampus. Since the results demonstrated in Fig. 4 are for a constant cortico-hippocampal coupling strength, system dynamics were stationary, allowing the use of these static measures with-out temporal windowing. Dynamic Synchrony Measure Dynamic measures of connectivity are well-suited for capturing evolving neural interactions in nonstationary data [ 111 ]. To quantify time-varying synchronization across cortical and hippocampal regions during simulated and empirical seizure data, we calculated a Dynamic Synchrony (DS) measure based on windowed, lagged cross-correlation. Given two bandpass-filtered time series x i ( t ) and x j ( t ), the instantaneous phase time series θ i ( t ) and θ j ( t ) were extracted using the Hilbert transform: where ℋ [·] denotes the analytic signal and arg(·) extracts the phase. For each electrode pair ( i, j ), DS was computed in 6-second windows with 50% overlap between windows. The windowed normalized cross-correlation at lag d was defined as: where and are the mean phases within window w . The DS value for window w was then defined as the maximum absolute cross-correlation within a lag range of ±1 s: where Δ is the maximum lag (1 s). This procedure produced a time series of DS values reflecting the dynamic synchrony between time series within a region and between regions. In simulated data, DS was computed directly on the eigenmode time series, while in empirical intracranial EEG (iEEG) recordings, bandpass filtering (1–15 Hz) preceded Hilbert transform and phase extraction. This approach accounts for potential phase lags and provides a consistent measure of large-scale synchronization across both simulated and empirical datasets. While traditional synchrony measures would suffice for non-pathological data due to its relatively stationary spectral content, the DS measure was applied uniformly across both pathological and non-pathological datasets to ensure methodological consistency. Spectral Content Analysis via Short-Time Fourier Transform To visualize the temporal evolution of spectral content during seizures, time–frequency spectrograms were calculated using the short-time Fourier transform (STFT). Individual electrode time series are analyzed using a Hanning window of 6 s duration with 50% overlap between consecutive windows and an FFT length of 6 s (sampling rate F s = 500 Hz). The resulting individual spectrograms were then averaged across all electrodes within a region to obtain a single spectrogram representative of cortical or hippocampal activity. This procedure was applied identically to empirical iEEG data and to the simulated cortical and hippocampal time series generated by the neural field modelling. AUTHOR CONTRIBUTIONS R.P. and M.B. conceived the study and developed the modelling framework. R. P. performed the simulations. P.A.R., R.P., and M.B. developed the theoretical and mathematical formalism. S.S. provided access to the de-identified iEEG data and R.P., S.S., and M.B. analyzed the data. J.D. provided the hippocampal surface meshes. R.P., M.B., J. R., J. P., S.S., A.B., N.K., J.D., J.M.S., P.A.R. and A.F. contributed to the interpretation of results and provided domain expertise. R.P. and M.B. drafted the initial version of the manuscript. All authors reviewed, edited, and approved the final version of the manuscript. COMPETING INTERESTS The authors declare no competing interests. DATA AND CODE AVAILABILITY The data used in this study consist of de-identified intracranial EEG seizure recordings obtained from a previously established clinical cohort at the Mater Advanced Epilepsy Unit, Mater Hospital, Brisbane. These data were originally collected for clinical purposes and made available for this study by co-author S.S. Due to ethics and consent restrictions, this data is not publicly available. Derived data supporting the findings of this study, including processed simulation outputs, are available from the corresponding author upon reasonable request. Custom MATLAB and EEGLAB scripts used for data analysis, as well as neural field simulation codes are available openly online [ 112 ]. ACKNOWLEDGEMENTS This work was supported by funding from multiple sources. M.B and R.P. were supported by the National Health and Medical Research Council (APP2008612). We also acknowledge the QIMR Berghofer Clinician Research Collaboration Grant Award 6626. J.C.P. acknowledges support from the Australian National Health and Medical Research Council (ID: 2034000) and Monash FMNHS Early Career Research Excellence Program. The authors acknowledge the facilities and scientific and technical assistance of the National Imaging Facility (NIF), a National Collaborative Research Infrastructure Strategy (NCRIS) capability, at the Hunter Medical Research Institute Imaging Centre, University of Newcastle. The funding bodies had no role in study design, analysis, decision to publish, or preparation of the manuscript. Funder Information Declared National Health and Medical Research Council , APP2008612 National Health and Medical Research Council , 2034000 National Imaging Facility, https://ror.org/01kdbsa35 Footnotes ↵ * Richa.Richa{at}newcastle.edu.au https://github.com/RichaPhogat/Cortico-HippocampalNFT Reference [1]. ↵ J. O’Keefe . The hippocampus as a cognitive map . Oxford University Press , 1978 . [2]. L. R. Squire and J. T. Wixted . 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