{"paper_id":"46bc608b-6bea-4459-b6c3-c2fd6c4bd738","body_text":"The computational bottleneck of basal ganglia output (and\nwhat to do about it)\nMark D. Humphries 1∗\n1. School of Psychology, University of Nottingham, Nottingham, UK.\n∗ Contact: mark.humphries@nottingham.ac.uk\nAbstract\nWhat the basal ganglia do is an oft-asked question; answers range from the selection\nof actions to the specification of movement to the estimation of time. Here I argue that\nhow the basal ganglia do what they do is a less-asked but equally important question.\nI show that the output regions of the basal ganglia create a stringent computational\nbottleneck, both structurally, because they have far fewer neurons than do their target\nregions, and dynamically, because of their tonic, inhibitory output. My proposed\nsolution to this bottleneck is that the activity of an output neuron is setting the weight\nof a basis function, a function defined by that neuron’s synaptic contacts. I illustrate\nhow this may work in practice, allowing basal ganglia output to shift cortical dynamics\nand control eye movements via the superior colliculus. This solution can account for\ntroubling issues in our understanding of the basal ganglia: why we see output neurons\nincreasing their activity during behaviour, rather than only decreasing as predicted\nby theories based on disinhibition, and why the output of the basal ganglia seems to\nhave so many codes squashed into such a tiny region of the brain.\nSignificance statement The basal ganglia are implicated in an extraordinary range of\nfunctions, from action selection to timing, and dysfunctions, from Parkinson’s disease to\nobsessive compulsive disorder. Yet however the basal ganglia cause these functions and\ndysfunctions they must do so through a group of neurons that are dwarfed in number by\nboth their inputs and their output targets. Here I lay out this bottleneck problem for\nbasal ganglia computation, and propose a solution to how their outputs can control their\nmany targets. That solution rethinks the output connections of the basal ganglia as a set\nof basis functions. In doing so, it provides explanations for previously troubling data on\nbasal ganglia output, and strong predictions for how that output controls its targets.\nIntroduction\nHow do the basal ganglia do any useful work? I will argue here that they suffer from a\nsevere computational bottleneck. Their output nuclei, through which they connect with\nthe rest of the brain, are markedly smaller than both their input sources and output tar-\ngets. Moreover, the standard view is that the basal ganglia’s output nuclei encode by\ndisinhibition, by the cessation of their inhibitory output (Chevalier et al., 1985; Chevalier\nand Deniau, 1990; Hikosaka et al., 2000; Basso and Sommer, 2011), which provides lim-\nited capacity for carrying information. Yet the moment-to-moment dynamics of the basal\n1\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n2\nganglia are implicated in a lengthy list of proposed functions, including action selection\n(Redgrave et al., 1999; Klaus et al., 2019), motor program selection (Mink, 1996), kine-\nmatic gain control (Turner and Desmurget, 2010; Park et al., 2020), perceptual decision\nmaking (Ding and Gold, 2013; Yartsev et al., 2018), duration estimation (Buhusi and\nMeck, 2005; Gouvˆ ea et al., 2015; Mello et al., 2015; Monteiro et al., 2023), signal routing\n(Stocco et al., 2010), and more. How one or more such complex functions are enacted\nthrough an output signal that has limited capacity in both size and dynamics is unclear.\nI will begin here by defining this computational bottleneck problem, first detailing\nthe anatomical expansion between the basal ganglia output nuclei and their targets, then\narguing that the disinhibition view of basal ganglia output is limited. This sets up two\nfundamental problems for the basal ganglia output: one, how does it re-expand? And,\ntwo, what dynamics does it use to code?\nI propose a solution to both these size and coding problems: that the basal ganglia\noutput neuron’s projections to their targets are a set of basis functions, and the output\nneurons’ activity sets the weights of those functions. All of these ideas will be elaborated\nbelow. This solution explains both how basal ganglia output can expand to the same\nscale as its targets, and why it would need to both decrease and increase its activity. It\ncan also account for troubling features of basal ganglia output, including why it has so\nmany apparently different coding schemes. Consequently, it is a step towards reconciling\nthe basal ganglia’s many apparent functional roles and may shed further light on why\ndysfunction of the basal ganglia is implicated in so many neural disorders.\nThe computational bottleneck problem\nThe structural bottleneck\nIn rodents, the basal ganglia output nuclei are traditionally considered to be the subtantia\nnigra pars reticulata (SNr) and entopeduncular nucleus (EP). In primates, the latter is\nequivalent to the internal segment of the globus pallidus (GPi). Regardless of their names,\nthese all share common anatomical properties: they receive input from the striatum and\noutput to structures including multiple regions of the thalamus, the superior colliculus,\nand the upper brainstem (Deniau and Chevalier, 1992; McElvain et al., 2021).\nThe striatum dwarfs the output nuclei. In rats, the striatum in one hemisphere contains\naround 2.8 million neurons, whereas the SNr and EP combined contain around 30,000\n(Oorschot, 1996), smaller by a factor of 100. In mice, the striatum contains about 400,000\nprojection neurons and the SNr around 12,000 neurons (numbers from the Blue Brain\nProject Cell Atlas, Rodarie et al., 2022); assuming that half of all projection neurons are\nD1-expressing and so project to the SNr, this gives a ratio of about 16:1 striatal projection\nneurons projecting to every SNr neuron in mice. This convergence of striatal projection\nneurons onto the basal ganglia output nuclei is well known, but we know little about the\nsizes of the target regions of the output nuclei.\nTo better understand the scale of the structural bottleneck, I used data from the\nAllen Mouse Brain Connectivity Atlas (Oh et al., 2014) to first identify a complete set\nof projection targets of the mouse SNr. The Atlas contains six experiments in which a\nfluorescent anterograde tracer was injected into the right SNr and filled at least 20% of its\nvolume. For each experiment I found which of a set of 295 non-overlapping target brain\nregions had evidence of projections from the SNr, by checking if the density of tracer in\nthat region exceeded some threshold. A threshold was necessary to eliminate image noise\nand other artefacts: without one, all 295 regions contained fluorescent pixels, implausibly\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n3\nimplying the SNr projects to every area of grey matter in the mouse brain, from medulla\nto olfactory bulb (Methods). The number of neurons in each retained target region was\nfound from the Blue Brain Project’s Cell Atlas for the mouse brain (Rodarie et al., 2022).\nThe total number of neurons in SNr target regions scaled with the size of the tracer’s\ninjection volume (Figure 1a). All injection volumes were smaller than the volume of the\nmouse’s SNr. Fitting a linear model to the scaling let me extrapolate to the number of\nneurons targeted by the whole SNr (Figure 1a, grey lines), and so estimate the ratio of\ntarget neurons to SNr neurons. This ratio fell to a stable value with increasingly stringent\nthresholds for eliminating noise (Figure 1c), estimated as 154:1.\nThe extrapolation to the whole SNr’s projection was based on three experiments that\nhad less than 40% of their injection volume inside the SNr (black symbols in Figure\n1a), potentially including neighbouring regions of the SNr that have different connection\npatterns. Weighting the linear model fit by the proportion of the injection inside the SNr\ngave practically identical ratios of target to SNr neurons (not shown). Estimating the\nnumber of target neurons directly for each of the three tracer injections almost wholly\nwithin the SNr (non-black symbols in Figure 1a), by extrapolating from the volume of\ntheir injection (Methods), also resulted in similar and stable ratios for the total number\nof target neurons to the number of SNr neurons (Figure 1e).\nThe total number of neurons in the target regions is an upper bound on the number\nof connections made by SNr neurons. To estimate a lower bound, I approximated the\narborisation of the axons from SNr in the target region by the volume density of the tracer\nin that region, scaling the number of neurons in each target region by the proportion of its\nvolume occupied by the tracer (Methods). This lower bound estimate of target neurons\nalso scaled with the tracer’s injection volume (Figure 1b), reached a stable value with\nincreasing noise threshold (Figure 1e), and was robust to alternative calculation using the\nthree within-SNr experiments (Figure 1f).\nThe estimated expansion from the basal ganglia output nucleus SNr to its targets\nranges from about 1:154 down to 1:13 (Figure 1). Even the lower bound on this expansion\nis thus about as large as the compression of inputs from striatum: the basal ganglia’s\noutput is then a considerable bottleneck, compressing its inputs by at least 10:1, and\nre-expanding them in its output targets by at least 1:10 (Figure 1g-h).\nThe dynamic bottleneck (or, why not disinhibition)\nBasal ganglia output neurons are constantly active. In rodents, they typically average 30\nspikes/s; in primates, around 60 spikes/s. They are also all GABAergic. This constant\nstream of high-frequency GABA release on to their target neurons has naturally led to\nthe assumption that they constantly inhibit their targets. From that has followed the\ndisinhibition hypothesis (Chevalier et al., 1985; Deniau and Chevalier, 1985; Chevalier\nand Deniau, 1990) that releasing this inhibition is key to how the basal ganglia encode\ninformation, by allowing their target neurons to respond to their inputs. This signalling by\ndisinhibition is the basis for most prominent conceptual (Mink, 1996; Redgrave et al., 1999;\nHikosaka et al., 2000) and computational (Gurney et al., 2001; Frank, 2005; Humphries\net al., 2006; Leblois et al., 2006; Bogacz and Gurney, 2007; Vitay and Hamker, 2010;\nLi´ enard and Girard, 2014; Lindahl and Hellgren Kotaleski, 2016; Dunovan et al., 2019)\nmodels of the basal ganglia.\nYet it is a myopic view of basal ganglia output that places a strong limitation on\nthe potential dynamic range of the output neurons, allowing coding by only a decrease\nin activity, and then often reduced to just whether the activity is on or off, a binary\nsignal (e.g. Hikosaka et al., 2000). There is no role for increases in activity, or changes to\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n4\n0 0.05\nDensity threshold\n0\n2000\n4000\n6000Target:SNr ratio 154\n0 0.05\nDensity threshold\n0\n50\n100Target:SNr ratio\n17\n0 0.05\nDensity threshold\n0\n50\n100\n150Target:SNr ratio 13\na b\nSNr SNr\nUpper bound Lower bound\nstriatum targets\nSNr\nstriatum\ntargets\nSNr\nc d\ne f\ng h\nFigure 1: The structural bottleneck of basal ganglia output in the mouse. a The total\nnumber of neurons in SNr target regions scales with the volume of tracer injection. Each symbol\nis an estimate from one tracing experiment of the Allen Mouse Brain Atlas; non-black symbols are\nexperiments with more than 90% of the injection within SNr. Red lines show linear fit and 95%\nconfidence interval. Grey line is the extrapolated total neurons in SNr targets from its volume.\nb As panel a, but estimating the targeted neurons in each region from the density of tracer in that\nregion’s volume (Methods).\nc The ratio of target neurons to SNr neurons for a range of thresholds on the minimum tracer\ndensity needed to include a target region. Data in panel a is for a threshold of 0.05. Number is\nthe asymptotic estimate of the ratio.\nd As panel c, but estimating the targeted neurons in each region from the density of tracer in that\nregion’s volume (Methods).\ne As panel c, for each of the three tracer experiments with injection volume confined to the SNr\n(non-black symbols in panel a). Here the expected number of neurons in SNr target regions is\ncomputed by scaling that experiment’s total number of target neurons by the proportion of SNr\nfilled by the injection. Number is the asymptotic estimate of the ratio averaged over the three\nexperiments.\nf As panel e, but estimating the targeted neurons in each region from the density of tracer in that\nregion’s volume.\ng Schematic of the upper bound of the SNr bottleneck. Target size from panel e; striatal D1R\npopulation estimated at 200,000 neurons (see text).\nh As for panel g, for the lower bound of the SNr bottleneck, target size from panel f.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n5\nthe patterns of activity (there are other theories: for example, there is evidence that the\noutput of the basal ganglia in songbirds controls the timing of activity in their thalamic\ntargets Goldberg et al. 2013). This compounds the structural bottleneck by then further\nlimiting how each neuron can send information. For example, taking the binary on/off\nsignal literally, disinhibition reduces the information coding capacity of the whole basal\nganglia output to just one bit per neuron, a few thousand bits in total.\nA solution: basal ganglia outputs are dynamic weights\nThe basal ganglia’s computational bottleneck problem is, then, that we have a limited\nnumber of free parameters – the output neurons and their dynamical range – compared\nto the number of outputs that we need to control. My proposed solution to this problem\nis a reframing of what the basal ganglia output encodes.\nI propose that the basal ganglia’s output connections are best understood as basis\nfunctions, and the level of basal ganglia output activity sets the weights on those functions.\nLet’s unpack those ideas, starting with a definition of basis functions.\nBasis functions are a typical solution for how to use a few parameters to control a\nlarge range of output. Figure 2a shows the key ideas. We first tile the output range we\nwant to control with a set of basis functions, such as the five Gaussians in the example of\nFigure 2a. Each basis function has a single weight that sets its contribution, such as the\namplitude of a Gaussian (Figure 2a, middle). Summing basis functions of different weights\ncan then create many different output functions over a large range of outputs (Figure 2a,\nbottom). Basis functions thus create an expansion from a few controllable parameters –\nthe weights – to a much larger target space.\nNow consider that the connections of the basal ganglia output neurons will have a\ndistribution of strengths (Figure 2b, top), the strength of a connection between an output\nneuron and its target being the product of the number of synapses and the conductances\nof those synapses. The idea then is that this distribution of strengths defines the basis\nfunctions (Figure 2b, bottom).\nConsequently, the amplitude of basal ganglia output activity sets the weights of those\nbasis functions. Figure 2c shows two examples of what this would look like: a particular\nvector of basal ganglia output activity scales the basis functions created by the output\nstrengths; when summed at each target, the larger target region as a whole receives a\ncontinuous function of inhibition, specified by far fewer basal ganglia output neurons than\ntarget neurons.\nThis idea rests on just two assumptions: that connections of the output neurons have\na distribution of strengths, and that these distributions overlap. It seems vanishingly\nunlikely that a given output neuron has an identical effective influence on each of its\ntarget neurons, so a distribution of strengths seems reasonable. And because the output\nof the basal ganglia is topographically organised (Deniau and Chevalier, 1992; Hoover and\nStrick, 1993; Lee et al., 2020; Foster et al., 2021; McElvain et al., 2021), with adjacent\nneurons projecting to adjacent targets, we might also reasonably expect these distributions\nof strengths to overlap. Beyond that, I emphasise that the schematics in Figure 2 are for\nillustration, not theory: I’m not claiming that distributions of output strengths have to be\nsymmetric, nor that their “centres” are distributed equidistant from each other in some\ntopographic space, nor that the distributions have to be the same, nor that they have to\nfollow any specific basis function used in the literature (such as radial basis functions).\nRather, the theory proposed is perhaps best expressed as: basal ganglia output activity\nis a dynamic weight on some function defined by the strengths of the output connections.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n6\n2 4 6 8\nTarget\nWeighted\nSummed\na b\nTarget region\nBasal ganglia output c\n2 4 6 8\nTarget\nbaseline\n2 4 6 8\nTarget\nInhibition\n2 4 6 8\nTarget\n2 4 6 8\nTarget\nbaseline\n1 2 3\nOutput\n1 2 3\nOutput\nBasis functions\nBasis functions\nFigure 2: Basis functions and basal ganglia output .\na Schematic of basis functions. A range of values is tiled by a set of basis functions, here five\nGaussians (top). Each basis function’s contribution is controlled by a single weight: the middle\npanels show two different weightings. Summing these weighted basis functions creates a continuous\nfunction spanning the range of values (bottom), controlled by just five parameters, the weights on\nthe basis functions.\nb Basal ganglia output connections define basis functions. Top: idealised network showing dis-\ntribution of basis function strengths, fanning from the output nuclei to a larger target region;\ncolour intensity is proportional to strength. Bottom: the basis functions created by the connection\nstrengths.\nc Basal ganglia output activity parameterises the basis functions. Using the network in panel\nb, two examples of how basal ganglia output activity (top) scales each neuron’s basis function\n(middle), which when summed as the input to each target creates a continuous inhibition function\n(bottom). Grey line indicates baseline output activity (top) and consequent inhibition of targets\n(bottom).\nLet’s now state the most general form of the theory and derive some general predictions\nfrom it.\nThe general form of the theory and its predictions\nConsider that b basal ganglia output neurons project to a set of n target neurons. We\nhave already established that b < n, the structural bottleneck. The theory proposes that\nthe goal of basal ganglia output is to create a specific function of inhibition across those\nn target neurons, which we can describe in a n-dimensional vector f with entries fi ≤ 0.\nThe theory can thus be expressed as the linear system\nDa = f, (1)\nwhere a is the b-dimensional vector of basal ganglia output activity, and D is the n × b\nmatrix of connection weights from the basal ganglia output to the set of target neurons\n(Figure 3a). Their values are also constrained: ai ≥ 0 as neural activity cannot be\nnegative, and Dij ≤ 0 because basal ganglia output is inhibitory. Matrix D defines the\nbasis functions, one column per output neuron. For example, in the schematic model of\nFigure 2b each column of D is a shifted version of the same, symmetric basis function.\nThus the vector a of basal ganglia activity are the dynamic weights on the basis functions\nD that gives the target function f.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n7\n0 50 100\nBG output neurons\n100\n1050\nUnique states\na\nb\nTriple\nDisinhibition-only\nConstant\nD f\na =X\nBG outputBG weights\nInhibition\nfunction in\ntarget \nn x 1b x 1 n x b \na\nFigure 3: Linear system model of basal ganglia output.\na Schematic of the linear model in Eq. 1.\nb Scaling of the number of possible functions f defined by basal ganglia outputs. Each line give\nthe number of unique functions possible with that number b of output neurons. Different lines\ncorrespond to the different number of states each output neuron can meaningfully be in: constant\n(1); disinhibition (2: on/off); triple (3 b) is up/down/unchanged.\nPrediction of non-uniform inhibition from uniform output\nBasal ganglia output neurons a are tonically active, firing at rest at a rate of around 30\nspikes/s in rodents and 60 spikes/s in primates. That a has non-zero values means a\nnon-trivial target function f is always defined.\nHowever, many theories implicitly assume that this tonic firing necessarily means there\nis a uniform level of inhibition, such that all values of f are the same. This is implied by\ntheories that tonic inhibition defines the “no-go” or “off” signal for selecting responses,\nactions, or motor programs (Mink, 1996; Redgrave et al., 1999; Hikosaka et al., 2000).\nThe model in Eq. 1 shows this is only true if the rows of D have the same sum. But\nthis is unlikely as only the columns of D, being the projections of each output neuron,\nare defined by development and plasticity. Consequently, the model predicts that tonic\nactivity of a set of output neurons causes a non-uniform inhibition of their targets.\nPrediction of increased output activity\nTraditionally, it is the cessation of this tonic inhibition, the disinhibition, that has been\nthe basis for key conceptual (Mink, 1996; Redgrave et al., 1999; Hikosaka et al., 2000) and\ncomputational (Gurney et al., 2001; Frank, 2005; Humphries et al., 2006; Leblois et al.,\n2006; Bogacz and Gurney, 2007; Vitay and Hamker, 2010; Li´ enard and Girard, 2014;\nLindahl and Hellgren Kotaleski, 2016; Dunovan et al., 2019) models of the basal ganglia’s\nfunction.\nThe model offered in Eq. 1 places no constraints on the values of a around the base-\nline tonic activity. Rather, the tonic activity values of a define a default f from which\nbehaviourally-necessary changes to f must occur. This predicts that both increases as well\nas decreases in output neuron activity can change f when basal ganglia output must cause\nor influence some behavioural event.\nThis prediction is borne out by data. Basal ganglia output neuron activity does increase\nin many tasks (Gulley et al., 1999; Handel and Glimcher, 1999; Gulley et al., 2002; Sato\nand Hikosaka, 2002; Jin and Costa, 2010; Fan et al., 2012; Rossi et al., 2016; Schwab et al.,\n2020) and the increases are as equally time-locked to action as the decreases (Sato and\nHikosaka, 2002; Jin and Costa, 2010; Fan et al., 2012). In some reports output neuron\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n8\nactivity can seemingly continuously encode parameters both above and below the nominal\n“tonic” firing rate (Barter et al., 2015).\nAllowing for increased output activity increases the basal ganglia’s scope for control.\nRestricting ourselves to classic disinhibition allows just two output states, on and off. The\nnumber of possible unique output combinations is then 2 b (Figure 3b). But adding just\none more output state, the increase above the tonic level, makes the number of unique\ninput combinations 3 b (Figure 3b): at 100 output neurons, this triple state can achieve\nmore then 5 × 1047 unique dynamic weight combinations, and hence that many different\nfunctions of inhibition f. Consequently, even a small group of basal ganglia output neurons\ncould control a wide repertoire of states in its target structures.\nPrediction of low variability in output activity\nStating the theory as the linear system Eq. 1 lets us ask an interesting question: is basal\nganglia output degenerate? That is, in some behavioural event for which the basal ganglia\nare necessary, can different combinations of increases and decreases of output neuron\nactivity achieve the same behavioural effect?\nLet’s assume that the same behavioural effect means achieving the same target function\nf. Then we are asking how many solutions exist to Eq. 1 (Druckmann and Chklovskii,\n2012): how many different basal ganglia outputs a achieve the same target function f.\nA heterogeneous linear system like Eq. 1 can have no, one, or an infinite number of\nsuch solutions. As we are interested in events where the basal ganglia have a necessary\nrole, then by definition we are interested in the set of f that can be achieved by the basal\nganglia output given D. So there must be at least one solution a for a given, behaviourally-\nrelevant, target function f. But is there more than one?\nIt seems unlikely. This is because the matrix of connection weightsD is almost certainly\nfull rank, having no linearly dependent columns. For structured basis functions, where each\ncolumn of D is approximately a shifted version of the same function (like Figure 2a), we\ncan guarantee that D is full rank by construction (Methods). For random basis functions,\nselecting the values of D from a wide range of symmetric probability distributions would\nguarantee it was full rank (Rudelson and Vershynin, 2009). A linear system with full rank\nD has at most one solution. Thus, either the basal ganglia output connections have a\ngenetically defined low-rank structure or there is only one basal ganglia output a that can\nachieve a given target function of inhibition f.\nHaving exactly one solution a to Eq. (1) predicts that individual basal ganglia neurons\nwould show little variability between repeated behavioural events that need the same f. I\nam proposing here that the necessary solution to a is for each behavioural event, so the\npredicted time-scale of this variation is around the gross changes in firing rate time-locked\nto an event, not precise spike-timing.\nConversely, observing considerable variability in basal ganglia output activity between\nrepeated events would imply either that f is not the goal of basal ganglia output, so the\ntheory here is incorrect, or that the connections of the basal ganglia to their targets are\nlinearly dependent and so basal ganglia output is redundant. This in turn would imply\nstrong constraints on how basal ganglia output is wired, in order to achieve this linear\ndependence.\nI am unaware of convincing data either way. While there is a considerable literature\non single neuron activity in both rodent SNr (e.g. Gulley et al., 1999, 2002; Bryden et al.,\n2011; Fan et al., 2012) and primate GPi/SNr (e.g. Handel and Glimcher, 1999; Sato and\nHikosaka, 2002; Nevet et al., 2007; Sheth et al., 2011) during tasks, I am not aware of any\nthat have quantified the trial-to-trial variability in that activity during exact repetition\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n9\nof a behaviour for which the basal ganglia are necessary. Close examination of example\nraster plots of individual SNr neurons aligned to the onset of eye movements (e.g. Handel\nand Glimcher, 1999; Sato and Hikosaka, 2002) suggests that gross changes of activity\nare highly consistent between trials, unlike, say, the rate variation of individual cortical\nneurons between repeated sensory stimuli (Tolhurst et al., 1983).\nHow the solution could work in practice\nLet’s illustrate the idea of basal ganglia outputs as dynamic weights in two concrete\ninstantiations: the control of cortical state by basal ganglia output to thalamus; and the\ncontrol of superior colliculus’ coding of saccade target by its inputs from the SNr.\nBasal ganglia output control of the repertoire of cortical states\nThe basal ganglia’s output to the thalamus is the main focus of much theorising because of\nits potential to control the dynamics of the cortical targets of those thalamic regions (e.g.\nHumphries and Gurney, 2002; Frank, 2005; Dunovan et al., 2019; M¨ oller and Bogacz, 2019;\nAthalye et al., 2020; Logiaco et al., 2021). But the thalamic regions contain more neurons\nthan the basal ganglia output nuclei, and the thalamus in turn is dwarfed by the numbers\nof cortical neurons (consider, for example, that of all the synapses arriving onto layer 4\nneurons in the visual cortex, thalamic synapses make up just 5% of the total Peters and\nPayne 1993). Here I illustrate how the idea of dynamic weights defined by the combinations\nof a few basal ganglia output neurons can allow control of cortical dynamics. In general,\nthe problem of how a few inputs can drives the states of a larger dynamical system is\nstudied under the control theory concept of controllability (Sontag and Sussmann, 1997;\nLiu et al., 2011; Gu et al., 2015; Kao and Hennequin, 2019); an interesting extension of the\nwork here would use controllability approaches to identify what form of target function f\nis ultimately necessary to control cortical states, and which elements of the cortical circuit\nmust be targeted to do so.\nLet’s consider a recurrent neural network (RNN) to model a region of cortex, as these\nnicely capture the basic problem: a network of mixed excitatory and inhibitory neurons\nthat is capable of producing complex dynamics (Figure 4a). A brief step in input to\nthis network produces a population response (Figure 4b, top). We can characterise this\npopulation response by the trajectory it creates in a low-dimensional space (Figure 4b, bot-\ntom). Such trajectories of neural activity in cortex correspond to specific arm movements\n(Churchland et al., 2012; Gallego et al., 2017; Rodriguez et al., 2024), elapsed durations\n(Voitov and Mrsic-Flogel, 2022), or choices (Harvey et al., 2012).\nA small fraction of these inputs, 10%, are from the thalamus. These thalamic inputs\nare in turn controlled by a small handful of basal ganglia output neurons, whose outputs\ncreate a set of basis functions to control thalamic activity. These basis functions are\nsymmetric, overlapping, and tiled in a ring as shown in Figure 4a (inset). The Appendix\ndiscusses the constraints on the number of thalamic neurons and basal ganglia weight\ndistributions implied by this model.\nDespite the small number of basal ganglia outputs they are sufficient to qualitatively\nchange the state of the cortical circuit. With all other inputs held the same, different\nvectors of basal ganglia output create different trajectories of activity in cortex (Figure\n4c). They do so by creating different functions of inhibition in the thalamus, defined by\nboth decreases (disinhibition) and increases in basal ganglia output activity. Crucially, we\nsee that these changes in trajectory would alter cortical coding (of an arm movement, a\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n10\na\nPC2\nPC3\nc d\nThalamus\nCortex\nBG output\nPC2\nPC3\nb\nFrom BGTo thalamus\nFigure 4: Basal ganglia output control of cortical dynamics .\na Schematic of the recurrent neural network (RNN) model, its inputs from other cortical regions\n(arrows) and its inputs from thalamus. Basal ganglia output to thalamus is a set of overlapping\nsymmetric basis functions (inset; grey-scale indicates strength of connection, white indicates no\nconnection). Thalamus projects to 10% of the cortical RNN units. Example simulations use 5\nbasal ganglia outputs, 20 thalamic units, and 200 cortical units.\nb Example response of all RNN units to stepped input (top), and projection of that RNN activity\ninto a low-dimensional space (PC: principal component). The trajectory of low-dimensional activity\ncaptures the move away from and return to baseline activity (black dot).\nc Trajectories of RNN activity in response to different basal ganglia outputs. Each line plots\nthe trajectory in response to a different basal ganglia output vector, with all other inputs held\nconstant. Output vectors were sampled from a uniform distribution centered on tonic activity,\nmodelling both increases and decreases of output.\nd Variation in basal ganglia output maps to variation in the trajectories of RNN activity. ρ:\nSpearman’s rank.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n11\ndelay, a choice) without need for that coding in the basal ganglia output.\nWork in songbirds has argued that basal ganglia activity is necessary to explore the\nrepertoire of potential movements, including the variation of syllable generation in songs\n( ¨Olveczky et al., 2005; Singh Alvarado et al., 2021). The dynamic weights view is also\nconsistent with this argument: the trajectories of activity in the cortical circuit vary in\nproportion to the distance between the output vectors of the basal ganglia (Figure 4d). So\ngreater variation in basal ganglia output could map to greater exploration of the cortical\nrepertoire of dynamics, and thus to motor behaviour or choices (de A Marcelino et al.,\n2023).\nHow output neurons can control superior collicular activity to influence\nthe orientation of the eyes and body\nAnother major target of basal ganglia output is the intermediate layers of the superior\ncolliculus. This structure plays a key role in orienting the eyes and body (Hikosaka et al.,\n2000; Felsen and Mainen, 2008; Villalobos and Basso, 2022), with activity in its inter-\nmediate layer acting as a command signal for eye movements to a particular location\n(Hikosaka et al., 2000). Much ink has thus been spilt on how the inhibitory signals ema-\nnating from the basal ganglia to the superior colliculus may in turn control the direction\nof gaze (Hikosaka and Wurtz, 1985; Jiang et al., 2003; Girard and Berthoz, 2005; Basso\nand Sommer, 2011).\nMost theories agree on the following (Hikosaka et al., 2000; Jiang et al., 2003; Girard\nand Berthoz, 2005; Chambers et al., 2011; Thurat et al., 2015). The intermediate, or\nmotor, layer of the superior colliculus represent the direction of gaze in two-dimensional\nretinotopic co-ordinates. For convenience we’ll consider them as a Cartesian grid of (x,y)\npositions in a two-dimensional plane. Neural activity in the intermediate layer thus rep-\nresents a motor command to direct gaze towards the location represented by the active\nneurons (Lee et al., 1988; Anderson et al., 1998). Basal ganglia inhibition of these collicular\nneurons is able to suppress changes in gaze direction to remembered locations (Mahamed\net al., 2014) and possibly stimulus-driven locations. Consequently, a pause in basal ganglia\noutput directed at collicular neurons representing a particular location allows the change\nin gaze to happen (Hikosaka and Wurtz, 1985).\nLess clear is how the basal ganglia output can provide that fine control over neural ac-\ntivity in the colliculus. The straightforward solution (Dominey and Arbib, 1992; Dominey\net al., 1995; Jiang et al., 2003; Girard and Berthoz, 2005) is that the basal ganglia output\nalso has a two-dimensional retinotopic map, and hence provides point-to-point control\nover collicular activity (Fig. 5a, top). But this scheme scales poorly because the number\nof possible co-ordinates scales linearly with the number of neurons (Fig. 5b, blue). And it\nseems at odds with the few neurons involved: of the basal ganglia output nuclei, only the\nsubstantia nigra pars reticulata (SNr) projects to the superior colliculus; that projection\noriginates from at most two-thirds of the SNr (Deniau and Chevalier, 1992; McElvain\net al., 2021); and even within that region the SNr neurons projecting to the superior col-\nliculus are potentially in the minority, as antidromic stimulation of the colliculus activates\nfar less than half of all sampled SNr neurons (Hikosaka and Wurtz, 1983c; Jiang et al.,\n2003).\nThe basis function idea provides a different solution, of the output neurons defining\nweights on basis functions tiling the two-dimensional plane. I give one example of how\nthis solution might work; others are possible. In this spanning code, the projection of\neach basal ganglia output neuron (or group of) is a basis function that spans one row\nor one column of the two-dimensional co-ordinates for gaze direction. Then a particular\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n12\n10 20\n5\n10\n15\n20\n0\n5\n10\n10 20\n5\n10\n15\n20\n0\n20\n40\n10 20\n5\n10\n15\n20\n0\n20\n40\n10 20\n5\n10\n15\n20\n-60\n-40\n-20\n0\n10 20\n5\n10\n15\n20\n0\n5\n10\na c\nd\nSuperior colliculus b\nSuperior colliculusExternal input\nBG output\nx\ny\nBG output\nx\ny\n0 50\nNumber of SNr neurons\n0\n200\n400\n600\n800Number of (x,y) co-ordinates\n 0 0.5 1\nRatio of x:y\n0\n200\n400\n600\n800Number of (x,y) co-ordinates\ne\nBG input\nf\ng\n10 20\n5\n10\n15\n20\n0\n20\n40\nx\ny\n10 20\n5\n10\n15\n20\n-60\n-40\n-20\n0\n10 20\n5\n10\n15\n20\n0\n20\n40\n10 20\n5\n10\n15\n20\n-60\n-40\n-20\n0\n10 20\n5\n10\n15\n20\n0\n5\n10\n10 20\n5\n10\n15\n20\n-60\n-40\n-20\n0\n10 20\n5\n10\n15\n20\n0\n5\n10\nBasis function\nTopographic\nBasis function\nTopographic\nFigure 5: Basal ganglia output control of superior colliculus .\na Potential schemes for basal ganglia inhibition of saccade targets in the intermediate layers of\nsuperior colliculus. The black circles are collicular neurons representing saccade targets in Cartesian\nco-ordinates. Top: topographic mapping of basal ganglia output to superior colliuclus, one output\nneuron per co-ordinate (blue). Bottom: a spanning code created by basis functions, with one\noutput neuron’s projection spanning one row or column (red).\nb Scaling of the number of controllable Cartesian co-ordinates by topographic or basis function\nmapping of basal ganglia output to superior colliculus. Topographic mapping is the best-case\nscenario of 1 neuron per co-ordinate.\nc Scaling of controllable co-ordinates with grid asymmetry. For a fixed number (50) of basal ganglia\noutput neurons, the scaling of the number of controllable co-ordinates as the superior colliculus\ngrid moves from a single row to a symmetric grid.\nd Simulations of basis function control of saccadic activity in superior colliculus. External input\nvia the superficial layers specifies a saccade target in Cartesian co-ordinates (left), input to a 20\n× 20 grid of collicular neurons middle. Twenty basal ganglia output neurons per side provide\ntonic inhibition of the superior colliculus: we plot here the inhibition received by each superior\ncolliculus neuron (right). Unchanging inhibition prevents the build-up of saccadic activity at the\ntarget location (middle).\ne As for panel d, but now the basal ganglia neurons whose basis functions include the x-coordinates\nor the y-coordinates of the target location pause their firing, thus allowing at their intersection\n(right) the build up of saccadic activity (middle).\nf Two competing external inputs (left) could cause saccadic activity to increase at both locations\nin colliculus (middle), even if only a few basal ganglia output neurons paused their firing (right),\nbecause the second, upper target falls in the column covered by paused basal ganglia neurons whose\nbasis functions are the y-coordinates.\ng As for panel f, but with other basal ganglia neurons increasing their firing and hence inhibition of\ncollicular neurons (right, darkest rows), thus suppressing the build-up of saccadic activity (middle)\nat the second target location.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n13\nco-ordinate is specified by the overlapping output of just two neurons (Fig. 5a, bottom).\nThe spanning code scales well, with the number of controllable co-ordinates rising\nquadratically with the number of neurons (Fig. 5b, red). As the spanning code uses\njust two output neurons to signal a particular location, its scaling is slower than a purely\ncombinatorial code (Fig. 3b), and its scaling is slower when the grid is asymmetric (Fig.\n5c). But it still scales better than standard theories of basal ganglia output to superior\ncolliculus: for a given number of output neurons there are always more controllable co-\nordinates for the spanning code than for the point-to-point wiring of a topographic map\n(Fig. 5b-c). So let’s check that the spanning code can indeed control collicular activity to\nprovide appropriate motor commands for gaze direction.\nImagine a model where input specifying the target gaze direction (from e.g. the frontal\neye fields or the superficial layer of the superior colliculus) arrives at a grid of intermediate\nlayer collicular neurons (Fig 5d; Methods). Activity at a location on that grid would\nrepresent a motor command to shift gaze to that target. At the same time, these neurons\nreceive constant inhibitory input from a set of basal ganglia neurons, each of whose output\nspans rows or columns of the collicular grid as in Figure 5a. This constant inhibition\nsuppresses all response to the target input (Fig 5d), preventing a shift in gaze direction.\nDropping the activity of basal ganglia output neurons whose projections intersect at the\ntarget location results in a hill of activity in the intermediate layer in neurons representing\nthat location (Fig 5e). Basis functions can thus allow suppression and selection of gaze\ndirection changes.\nThis selection requires only a decrease in basal ganglia output, but I have been arguing\nthat they encode bidirectional “dynamic weights”: what then might an increase in basal\nganglia output encode here? One answer is to correct for unwanted loss of inhibition\nelsewhere on the two-dimensional map of gaze directions. Consider two competing target\nlocations that lie on the same column (Fig 5f, left): pausing intersecting basal ganglia\noutputs for one target could now result in a hill of activity at both target locations on the\nsuperior colliculus’ map (Fig 5f, implying two simultaneous but different changes in gaze\ndirection. However, increasing the output of basal ganglia neurons whose basis functions\nare the corresponding row of the unwanted target location will suppress the hill of activity\nat this location (Fig 5g). Increasing basal ganglia output could ensure that at most one\ntarget location for gaze direction becomes active in the intermediate superior colliculus.\nDiscussion\nBasal ganglia output neurons are vastly outnumbered by their target neurons in the tha-\nlamus and brainstem. To this structural bottleneck is added the further dynamical bottle-\nneck that this output is both constant and inhibitory. It is unclear how the basal ganglia’s\noutput re-expands, both structurally and dynamically, to provide suitable control over its\ntarget regions.\nI’ve offered as a solution the following idea: the activity of a basal ganglia output\nneuron defines the weight on a basis function defined by the pattern of its connections to\nits target regions. In this way, basal ganglia output can re-expand, for arbitrarily complex\nfunctions can in principle be constructed from the summation of a set of basis functions.\nThe complex function approximated in this model is the level of inhibition across the\ntarget neurons of the basal ganglia output.\nThis model makes three predictions. First, output neurons all in their tonic state of\nconstant firing do not necessarily imply uniform inhibition of their targets. Second, that\nboth decreases and increases of output neuron firing are necessary to create the target\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n14\nfunction. Third, that individual output neurons will show low variation in their activity\nbetween identical events.\nImplications for our understanding of the basal ganglia\nThere are three main implications of these ideas and their predictions.\nThe first is that disinhibition is not the key to how basal ganglia output operates.\nWeights need to change in both directions to control basis functions, and create the desired\ntarget function by their summation. So under this account we’d expect basal ganglia\noutput to both increase or decrease as necessary to set the desired target function of\ninhibition in its target. This provides a functional explanation for why basal ganglia\noutput activity increases as well as decreases during movement or action (Gulley et al.,\n1999, 2002; Jin and Costa, 2010; Fan et al., 2012; Barter et al., 2015; Rossi et al., 2016;\nSchwab et al., 2020), which cannot be accounted for by classic disinhibition-based theories.\nThe second is that we can think of basal ganglia output as allowing rapid exploration of\ndifferent activity patterns in a target region, which could be crucial in learning; for example\nof new actions such as skilled limb movements or songs. This is why I’ve called the basal\nganglia output “dynamic weights” throughout, as simply by changing their activity the\noutput neurons define a new function of inhibition in their target, which creates a different\nresponse in that target to the same input (Fig 4).\nThe third is that basal ganglia outputs have no intrinsic “code”, but are control signals\nfor a target region. The SNr has many apparent codes. Changes in SNr activity align\nto the onset of changes in gaze direction in monkeys that are stimulus or memory-driven\n(Hikosaka and Wurtz, 1983a,b; Handel and Glimcher, 1999, 2000; Basso and Wurtz, 2002;\nSato and Hikosaka, 2002) and changes in movement in rodents (Lintz and Felsen, 2016).\nJin and Costa (2010) report that SNr activity changes at the beginning and end of an\naction sequence, not at each action (though such discrete stop and start coding has not\nbeen found in striatum Sales-Carbonell et al. 2018). By contrast, Rossi et al. (2016) report\nSNr activity changes align to each individual lick a mouse makes on a spout. Fan et al.\n(2012) demonstrate that a pause in SNr activity can last for as long as a mouse holds down\na lever is held, seemingly coding duration or a sustained action, rather than action onset or\noffset. And Barter et al. (2015) offer a startling demonstration of apparently continuous\ncoding of a mouse’s head position in the (x,y) plane by the activity of individual SNr\nneurons, which typically coded either the x- or y-axis displacement of the head from its\ncentral position. Worse, as they are the bottleneck between striatum and the rest of the\nbrain the basal ganglia output nuclei likely inherit other variables known to be encoded in\nstriatum, and there are many of those, including time (Gouvˆ ea et al., 2015; Mello et al.,\n2015), decision variables (Ding and Gold, 2013; Yartsev et al., 2018), possible actions\n(Klaus et al., 2019), their predicted value (Samejima et al. 2005; but see Elber-Dorozko\nand Loewenstein 2018), and their kinematics (Rueda-Orozco and Robbe, 2015; Yttri and\nDudman, 2016). How so few neurons could seemingly encode such a range of different\nvariables is unclear. A starting point for a solution could be the dynamic weights theory\noffered here: basal ganglia output is not coding these variables, but are control signals for\nencoded variables in their target regions.\nPredictions for effects in target brain regions\nThe interpretation of basal ganglia output as encoding “dynamic weights” is a general\nprinciple for how that output can do useful work in its many target regions of the brain.\nFurther specific predictions of this idea depend on the region targeted.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n15\nOne prediction is that this re-expansion allows the basal ganglia output to shift cortical\nactivity across an extensive repertoire of different states, via the shaping of thalamic output\nto cortex. The idea that basal ganglia output shifts the state or trajectory of cortical\nactivity has been gaining traction: Athalye et al. (2020) looked at the problem of how\nmotor cortical activity re-enters a desired state after a reward, and proposed a conceptual\nmodel for how plasticity at cortico-striatal synapses may alter basal ganglia output to shift\ncortical activity; Logiaco et al. (2021) looked at the problem of how motor cortex can drive\nsequences of movements, developing and analysing a model of how shifting basal ganglia\noutput can switch motor cortical activity states and thus create movement sequences\n(though based only on disinhibition setting thalamic units on or off). Which regions\nof cortex this prediction extends too is unclear; speculatively, it would be in the layers\n(II/III and V) of cortical regions that receive direct input from the motor and intralaminar\nthalamic nuclei that the basal ganglia output directly influences (Nishimura et al., 1997;\nKha et al., 2001; Middleton and Strick, 2002; Bodor et al., 2008; Kuramoto et al., 2011).\nThis prediction is consistent with data from primates (Sauerbrei et al., 2020), rodents\n(Inagaki et al., 2022) and songbirds (Moll et al., 2023) showing that motor thalamic input\nto motor cortex is necessary for initiating, or changing to, a discrete element of movement.\nIt follows that manipulating the state of basal ganglia output projections to thalamus\nshould change the state of cortical activity, and hence alter behaviour. Recent studies\nthat optogenetically manipulated SNr axon terminals in motor thalamus reported that in-\nhibiting these SNr terminals during a licking task biased licking to the contralateral side,\nwhereas activating these terminals stopped licking all together (Morrissette et al., 2019) or\nprevented impulsive licking (Catanese and Jaeger, 2021). These data are consistent with\nshifting the state of cortical activity; but they are also consistent with classic disinhibi-\ntion ideas that pauses in basal ganglia output are a go signal and increases in the output\nare a stop signal. Such optogenetic stimulation that broadly targets all terminals with a\nstereotyped pattern of stimulation is repeatedly setting all the dynamic weights to approx-\nimately the same values. Distinguishing classic disinhibition and dynamic weight accounts\nof basal ganglia output would instead need selective stimulation of non-overlapping sets\nof SNr terminals: the dynamic weights idea predicts this would result in different states\nof cortical activity, and hence potentially different behavioural responses.\nIf the basis functions defined by the connections of basal ganglia output neurons have\nsome topographic organisation, as in the example of superior colliculus, then different\npredictions arise for manipulating their activity. Selective optogenetic activation of a few\nSNr neurons that project to the superior colliculus would be predicted to maximise the\nweight on their basis functions, strongly inhibiting a region of the intermediate superior\ncolliculus, and so create a region of visual space that it is difficult to shift gaze or orient\ntowards. The shape of that region would depend on the shape of the basis functions. The\nspecific spanning code advanced here (Fig 5) would predict that the region of visual space\nwould cover entire x- or y- co-ordinates. This prediction is consistent with reports that\nthe activity of individual SNr neurons encodes the entire x-axis or y-axis displacement of\nthe head in mice (Barter et al., 2015). That said, as noted above the spanning code is not\nan optimal form of basis function tiling, and other tilings of the collicular representation\nof gaze direction, including random projections, would be worth further exploring.\nWhat the dynamic weights idea does not yet address\nThere are a few key issues that further development of this “dynamic weights” idea could\nusefully address. The first is its precise role in behaviour. Most current basal ganglia\ntheories focus on how its output controls action, but fall largely into two opposing camps.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n16\nOne is that basal ganglia output controls action selection, via pauses in its tonic inhibition\n(Redgrave et al., 1999; Bogacz and Gurney, 2007; Li´ enard and Girard, 2014; Klaus et al.,\n2019); the other is that basal ganglia output controls action specification, by its reduced\nactivity modulating some aspect of the kinematics or gain of movement (Turner and\nDesmurget, 2010; Park et al., 2020; Thura et al., 2022). The dynamic weights idea is\ncurrently agnostic to these theories: indeed, it suggests that coding is in the target region,\nnot in basal ganglia output per se. Changes in cortical state by basal ganglia output could\nbe the selection of action or change in the kinematics of an action. Similarly, the example\ngiven of superior colliculus control follows previous models of SNr output to colliculus\n(Girard and Berthoz, 2005) in interpreting the hill of activity as selecting an action, the\nnew direction of gaze; but one could interpret this as specifying action, setting the velocity\nof the change in gaze direction by controlling the size of the hill of activity.\nBoth canonical action selection (e.g. Gurney et al., 2001; Frank, 2005; Humphries et al.,\n2006; Li´ enard and Girard, 2014; Lindahl and Hellgren Kotaleski, 2016) and specification\n(e.g. Yttri and Dudman, 2016; Park et al., 2020) theories also assume basal ganglia output\nare organised as a collection of parallel populations or “channels”, each one representing\nan option to be selected or specified. While this assumption is the basis for powerful\nexplanations of how the basal ganglia circuit can implement selection or specification, it\nalso strongly restricts the coding capacity of the basal ganglia output. The theory here is\nas yet agnostic to the size of the basal ganglia output population a over which a target\nfunction f could be specified. Given the broadly overlapping targets of neighbouring SNr\nneurons (Deniau and Chevalier, 1992) there could be just one population whose goal is to\ncreate a specific trajectory of activity in its targets (W¨ arnberg and Kumar, 2023).\nA second issue is of how control of the dynamic weights can be learnt. The theory\noutlined here assumes the connections made by the basal ganglia output nuclei have fixed\nstrengths (Figure 2), defining the basis functions, and it is the activity of the output\nneurons that set the weights of those functions. Unlike traditional views of basis functions\nwith weights that are learnt, these dynamic weights can be adjusted on the fly by input\nfrom upstream nuclei of the basal ganglia. Learning when and how to adjust them becomes\na question of how feedback from the environment can change the input to the output nuclei.\nAn obvious answer is that environmental feedback is the prediction error signal conveyed\nby phasic dopamine release in the striatum (Schultz et al., 1997; Bayer and Glimcher,\n2005; Rutledge et al., 2010; Hart et al., 2014; Howe and Dombeck, 2016); and that this\nmodulates plasticity at cortico-striatal synapses (Reynolds et al., 2001; Shen et al., 2008;\nGurney et al., 2015) to change the input to the basal ganglia’s output nuclei. Extending\nthe theory here to include explicit striatal input and cortico-striatal plasticity would be\nan interesting line of work.\nThat striatally-targeted dopamine acts like a reward prediction error and modulates\ncortico-striatal plasticity are key reasons why the basal ganglia are often viewed as a central\npart of the brain’s distributed system for reinforcement learning (Ito and Doya, 2011).\nIn this view, the value of a state or action is represented in the cortex, cortico-striatal\nconnection weights, or striatal activity (Houk et al., 1995; Joel et al., 2002; Khamassi\net al., 2005; Samejima et al., 2005; Bogacz and Larsen, 2011; Collins and Frank, 2014;\nBlackwell and Doya, 2023) [see also Elber-Dorozko and Loewenstein (2018) for dissenting\nevidence], and the downstream basal ganglia circuit enact policy by selecting (Frank, 2005;\nHumphries et al., 2012; Collins and Frank, 2014) or specifying (Yttri and Dudman, 2016)\nan action based on that value. As reinforcement learning algorithms place no constraints\non how policy is implemented, this view is silent on how the basal ganglia circuit selects\nor specifies action, and silent on the existence of the bottleneck and of any solutions to\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n17\nit. Some forms of reinforcement learning use basis functions to approximate a continuous\nspace of states or actions (Sutton and Barto, 2018). This use of basis functions is distinct\nfrom the ideas offered in this paper: here, the basis functions are a property of anatomy,\nnot abstract quantities, and are not explicitly representing anything. Indeed, all theories\nmapping elements of reinforcement learning algorithms to the basal ganglia could turn out\nto be false, but if they were false it would not affect the ideas in this paper. The converse\nmay not hold: an open question is whether the re-expansion of basal ganglia output places\nany limits on mappings of reinforcement learning algorithms to the basal ganglia circuit.\nFinally, while the dynamic weights idea provides a solution to the problem of the\ncomputational bottleneck, it does not explain why this bottleneck exists. Some ideas\nseem worth pursuing here. Producing spikes makes up much of a neuron’s energy demand\n(Attwell and Laughlin, 2001; Laughlin, 2001; Sengupta et al., 2010), and so if the high\nactivity rates of the output nuclei are necessary to their function, this could place energetic\nconstraints on their size. Such bottlenecks can also occur as networks evolve when the\nmapping between their input and output has fewer dimensions than either of the input\nor the output itself (Friedlander et al., 2015). But, however it arose, the computational\nbottleneck of basal ganglia output provides a strong challenge to any theories for functions\nof the basal ganglia.\nAcknowledgements Though there is a single author’s name on the byline, these ideas\nrepresent the synthesis of work from many labs in the vibrant basal ganglia field. They\nparticularly benefited from feedback on my initial presentation of these ideas at the 2016\nGRC Basal Ganglia conference, for which I thank Nicole Calakos & Mark Bevan for the\ninvitation to speak, and from the attendees of the 2023 IBAGS meeting. I thank Tom\nGilbertson and Bhadra Santhi Kumar for comments on drafts.\nThis work was supported by the Medical Research Council [grant numbers MR/J008648/1,\nMR/P005659/1 and MR/S025944/1] and Innovate UK [grant number 10036282].\nMaterials and methods\nCode availability All MATLAB code is available under a MIT License from https:\n//github.com/mdhumphries/Basal_Ganglia_Bottleneck_simulations.\nEstimating the expansion of basal ganglia output\nTargets of the mouse SNr were accessed from the Allen Mouse Brain Atlas (Oh et al.,\n2014) at https://connectivity.brain-map.org/. The Atlas contains seven experiments\nthat had the injection of a fluorescent anterograde tracer targeted to the SNr of the\nright hemisphere. One experiment (ID:3035444) filled a volume (0.04mm 3) an order of\nmagnitude smaller than the others, occupying less than 3% of the SNr, and so was not\nused here. Table 1 gives data on the six retained experiments.\nEach experiment contained a complete list of all regions in the Allen Common Co-\nordinate Framework V3.0 that had tracer found in them, in both hemispheres, and the\nvolume and density of fluorescent pixels in each region. The Allen Common Coordinate\nFramework follows a hierarchy from the whole brain, through the grey matter, down to\nfine-grained subdivisions of nuclei: identified target regions thus overlapped with, or were\nsubdivisions of, others. To find a single, non-overlapping set of targets, I used the list of\n295 non-overlapping grey matter regions from Oh et al. (2014), their Supplementary Table\n1.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n18\nTable 1: Allen Mouse Brain Atlas experiments . The proportion of each injection that fell\nin the SNr was manually scraped from each experiment’s webpage at https://connectivity.\nbrain-map.org/. Proportion of SNr injected used the SNr volume of 0.69 mm 3 from the Blue\nBrain Cell Atlas.\nID Volume in-\njected (mm 3)\nProportion in\nSNr\nProportion of\nSNr injected\n478096249 0.49 0.41 0.30\n158914182 0.54 0.29 0.23\n478097069 0.43 0.32 0.21\n100141993 0.23 0.95 0.32\n299895444 0.36 0.94 0.51\n175263063 0.17 0.96 0.24\nFigure 6: The need for a threshold for noise in tracing experiments. Here I plot the total\nnumber of target neurons when including every target region of the SNr with detectable fluorescent\npixels in the Allen Mouse Brain Connectivity Atlas experiments. As all or almost all regions had\ndetectable pixels, the total number of neurons is at or close to the total number of neurons in the\nmouse brain. Also note how the number of neurons does not scale with the injected volume.\nA complete set of neuron numbers for each of those regions in the mouse brain was\nobtained from Blue Brain Cell Atlas dataset published in Rodarie et al. (2022), their S2\nExcel file.\nWith these data to hand, for each experiment I did the following:\n1. Found all targets within the list of 295 unique regions, in both hemispheres\n2. Rejected any target region that had a fluorescent pixel density less than some thresh-\nold θ\n3. Looked up the neuron count nt in each target region t from the Blue Brain Cell Atlas\n4. Computed the total potential SNr connections: summed neurons across all N re-\ntained target regions PN nt\n5. Computed an estimate of SNr connections by estimating axonal arborisation in each\ntarget: weighted the neuron count in each region by its density dt ∈ [0, 1) of fluores-\ncent tracer PN nt × dt.\nThresholding by the density of fluorescence within each target region (step 2) elim-\ninated image noise and other artefacts: without a threshold, all 295 regions contained\nfluorescent pixels implausibly suggesting the SNr projected to the entire brain (Figure 6).\nLinear models were fit to the scaling of the total number of target neurons (step 4)\nand of the estimate of SNr connections (step 5) with the volume of injection in each\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n19\nexperiment (fitlm, MATLAB2023a). Extrapolating these models to the volume of the\nSNr (predict, MATLAB2023a) gave upper and lower bounds on the total number of SNr\ntarget neurons. From these were computed upper and lower bounds on the ratio of target\nto SNr neurons. Computing the upper and lower ratio bounds for a range of density\nthreshold θ between 0 and 0.05 in steps of 0.005 revealed they reached an asymptote with\nincreasing θ (Figure 1c,d) implying a stable estimate. The asymptote itself was detected\nas three consecutive bound estimates that changed by less than 1, and was reported as\nthe final of these estimates.\nAs Table 1 shows, three of the experiments had injections that considerably exceeded\nthe bounds of the SNr. Robustness to this limitation was tested in two ways. In the first,\nthe linear model was fit weighting the data points by the proportion of SNr filled in each\nexperiment: this gave the same results for the asymptotic ratios as the unweighted linear\nmodel.\nIn the second, I separately considered each of the three experiments with more than\n90% of their injection within the SNr. For each experiment, I first computed the estimated\ntotal number of target neurons and the estimate of SNr connections (steps 4 and 5 above).\nI then computed the estimated number of SNr neurons ˆS within the injection volume,\ngiven by\nˆS = # neurons in SNr × Injection volume\nVolume of SNr , (2)\nand finally computed the ratio of target neurons to ˆS. The resulting upper and lower\nbound ratios of target to SNr neuron numbers also reached an asymptote for each of three\nexperiments (Figure 1e,f). The asymptote was computed as above separately for each\nexperiment; the mean of these is reported.\nUsing basis functions for expansion\nMy proposal here stems from the use of basis functions to approximate a target function,\nso I give a brief account of that here to make the analogy clear.\nThe target function is f(x), which can be of any dimension (for example, I use one-\ndimensional functions in Figures 2 and 4, and two-dimensional in Fig 5). In the most\ngeneral form of approximatingf(x), we choose some basis functiong(x, Ω), which is defined\non some interval x and takes parameters Ω, and we obtain the approximation f ′(x) of f(x)\nby\nf ′(x) =\nBX\ni=1\nwig(x, Ωi), (3)\nthe sum of B such basis functions each parameterised by Ω i.\nApproximating a function typically uses the radial basis functions g(x − c, σ), which\nare symmetric about their centre c and have a single parameter σ that defines their width.\nGaussians are one type of this class of functions, as shown in Figure 2a. The approximation\nof f(x) is then\nf ′(x) =\nBX\ni=1\nwig(x − ci, σ), (4)\nthe sum over a set of B such basis functions, placed at centres c1, c2, . . . , cB, and each\nwith its own weight wi. Approximating f(x) then depends on choosing g(x), setting an\nappropriate number B of these, and finding appropriate weights to minimise some error\nfunction e(f(x) − f ′(x)). When using basis functions to fit curves (in one dimension) or\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n20\nsurfaces (in two or more dimensions) to sampled data, typically the number and location\nof sampled data-points respectively determines B and the set of ci.\nExpansion occurs because we want the target function f(x) to be defined over n points\nin one dimension (or n × m points in two dimensions, and so on), and typically B ≪ n.\nThis is especially true when using basis functions to approximate functions across a full\nrange of n points using just B sampled data points.\nThe basal ganglia output as a set of dynamic weights\nThe theory outlined here is that g(x) is realised by the strength distribution of an output\nneuron and the weights wi are defined by the activity of that neuron (or equivalently, the\nstrength distribution of a group of output neurons taken collectively, and their collective\nactivity). Then f ′ is the inhibition function defining the input to every neuron in the\ntarget nucleus.\nTo make this more concrete, consider that there are b basal ganglia output neurons\nthat project to target region(s) containing n neurons, with b ≪ n. Then expressing the\nbasis function idea in matrix form, we get:\nDa = f , (5)\nwhere a is the b-length vector of basal ganglia output activity, f is the n length vector\nof inhibition arriving at the target neurons – the target function in Equation 4 – and D\nis the n × b matrix that defines the basis functions. For example, for a one-dimensional\nset of overlapping basis functions that define a ring (cf Figure 4a), each column of D is a\nbasis function, shifted circularly across columns.\nIf the columns of D are each a circularly shifted version of the same basis function v,\nand no column can be identical, then the construction of D is:\nD =\nbX\ni\nai ⊗ vi, (6)\nwhere vi is the ith shifted version of v and ai is a vector with a 1 at the ith entry and 0s\nat all others. Consequently, Eq 6 defines a full rank matrix, i.e. its columns are linearly\nindependent.\nRecurrent neural network modelling of cortex\nThe recurrent neural network model of cortex contained N = 200 units, 80% excitatory\nand 20% inhibitory, whose dynamics were given by:\nτ dx\ndt = −x + Wr(x) + I(t), (7)\nwhere x is the N-length vector of unit potentials, τ = 20 ms is the network’s time constant,\nW is the N × N connectivity matrix, r() is a function that defines each unit’s firing rate,\nand I(t) is an N-length vector of inputs to each unit at time t. I used here the stabilised\nsupra-linear network (Ahmadian et al., 2013; Rubin et al., 2015; Hennequin et al., 2018)\nin which the firing rate of the ith unit was given by the rectified power-law,\nr(xi) = k⌊xi⌋η, (8)\nwith k = 0.1 and η = 2. Numerical simulations used forward Euler with a time-step of\n0.1ms.\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n21\nRandom connectivity matrices W of size N × N were created by connecting units with\nprobability p = 0.1, then randomly assigning columns to be excitatory or inhibitory units.\nWeights of connections in a column were then defined asw0/\n√\nN for excitatory connections\nand (−γw0)/\np\n(N) for inhibitory connections, where w0 = R/\np\np(1 − p)(1 + γ2)/2, the E:I\nweight ratio γ = 3, and the spectral radius R = 10 (Rajan and Abbott, 2006; Hennequin\net al., 2014). I tested versions with and without further stability optimisation of the\nresulting random matrix W; further optimisation used the algorithm of (Hennequin et al.,\n2014) that minimises the maximum real eigenvalue of W, resulting in networks whose\ndynamics recapitulate those of motor cortex (Hennequin et al., 2014; Rodriguez et al.,\n2024). Similar results were obtained in both random and optimised W; results from a\nrandom network are plotted in Figure 4. Similar results were also obtained in networks\nwith a 50:50 ratio of excitatory and inhibitory units (not shown).\nThe vector I(t) comprised inputs from unmodelled regions of cortex and from thalamus.\nBackground input of I(t) = 0.1 was applied for t < 1000 and t > 2000. For time 1000 ≤\nt ≤ 2000 the inputs stepped up or down from background.\nThe stepped thalamic input to cortex was assigned to n = 0.1N randomly chosen\nentries of I(t). It was modelled as the vector T whose n entries were given by T =\nTb − Da, where Tb = 1 was baseline thalamic activity, a was the b = 0.25t length vector\nof basal ganglia output activity randomly chosen in [0 , 0.4], and D was the n × b matrix\nof connection weights from basal ganglia output to thalamus. Each column of D was thus\nthe basis function defined by each basal ganglia output neuron’s projections to thalamus:\nhere each basis function was symmetric with 1 at the centre and steps down of 0.2 on\neither side; the basis functions were circularly shifted to uniformly tile the matrix (Figure\n4a).\nThe stepped cortical input was modelled by the remaining 0 .9N entries in I(t), which\nwere randomly assigned an initial value in [0 , 1]. The stepped thalamic and cortical input\nwere then held constant for 1000 ≤ t ≤ 2000.\nWhen simulating the effects of changing basal ganglia output, only the entries of a\nwere resampled in each simulation, with all other stepped inputs I(t) sampled once and\nthen held constant across simulations.\nSuperior colliculus grid model\nThe build-up layer of the superior colliculus was modelled as a N × N grid of units, with\nN = 20 here. The dynamics of units on the grid were given by\nτ ds\ndt = −s + u(t) + Da, (9)\nwhere s was the N × N length vector of collicular unit activity and τ = 10 ms was the\nunit’s time constant. Inputs specifying the saccadic target location were represented in\nthe N × N-length vector u. Input from b = 2N basal ganglia output neurons was defined\nby the b-length vector of their output a and the (N × N) × b matrix D that defined the\nbasis functions. The entries of D were -1 or 0, set so that each output neuron connected\nto either an entire row or column of the collicular grid (Figure 5a, bottom).\nColliculus output activity plotted in Figure 5 is given by the rectified linear function\nr = max(0, s).\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n22\nAppendix: Limits on the number of thalamic neurons reach-\nable by basal ganglia output\nThe model of basal ganglia output to thalamus adopted in the main text (Figure 2) gives\nsome interesting insights into potential constraints on the divergence of basal ganglia\nprojections to thalamus.\nIn the model, each of the b basal ganglia output neurons has a symmetric distribution\nof output strengths (main text Figure 2a, inset), which defines its basis function. Given\nthe number of possible discrete strength values Nb, the width of this distribution is NT =\n2Nb − 1, which defines the number of thalamic neurons reachable by one basal ganglia\noutput neuron.\nEach basal ganglia output strength distribution is centred on a specific thalamic neuron\nti. Let’s assume the distributions tile the thalamus in a one-dimensional ring to avoid\nedge effects, and the centres of the distributions are spaced the same number d of thalamic\nneurons apart. The number of possible thalamic neurons in this ring model is thenT = d·b.\nIf we want every thalamic neuron to receive basal ganglia input, then the maximum\ndistance d between the centres of the distributions must also be the width of the distribu-\ntions: dmax = 2Nb − 1. Consequently, the maximum number of thalamic neurons must\nbe Tmax = (2Nb − 1)b.\nBut if the basal ganglia output distributions are spaced at distance dmax then each\nthalamic neuron receives only one basal ganglia input, and so the basis functions are\nnot overlapping. The distance between basal ganglia output distributions must therefore\nbe d < d max. How much smaller would depend on the minimum desired number of\noverlapping basal ganglia inputs to each thalamic neuron.\nTable 2 gives the minimum number of overlapping basal ganglia output neurons in\nthis ring model, for a range of distances d between, and resolutions Nb of, the basal\nganglia output strength distributions. For example, with the centres of the distributions\nspaced d = 3 thalamic neurons apart, and the distributions having NS = 6 possible\nstrength values, each thalamic neuron receives input from at least three basal ganglia\noutput neurons.\nTable 2: Minimum number of basal ganglia inputs to one thalamic neuron . Each entry\nis the minimum number of inputs each thalamic neuron receives at the specified strength resolution\n(Nb) and distance d between strength distributions. These were numerically enumerated, using\nb = 2 · max Nb.\nd N b\n3 4 5 6 7 8 9 10 11 12 13 14 15 16\n1 5 7 9 11 13 15 17 19 21 23 25 27 29 31\n2 2 3 4 5 6 7 8 9 10 11 12 13 14 15\n3 1 2 3 3 4 5 5 6 7 7 8 9 9 10\n4 1 1 2 2 3 3 4 4 5 5 6 6 7 7\n5 1 1 1 2 2 3 3 3 4 4 5 5 5 6\n6 0 1 1 1 2 2 2 3 3 3 4 4 4 5\n7 0 1 1 1 1 2 2 2 3 3 3 3 4 4\n8 0 0 1 1 1 1 2 2 2 2 3 3 3 3\n9 0 0 1 1 1 1 1 2 2 2 2 3 3 3\n10 0 0 0 1 1 1 1 1 2 2 2 2 2 3\nWe see that the minimum number of inputs falls rapidly with increasing distance\n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted February 25, 2025. ; https://doi.org/10.1101/2024.10.23.619790doi: bioRxiv preprint \n\n23\nbetween distributions. If the distance is greater than 4, then increasing the minimum\nnumber of inputs above 2 or 3 requires a large increase in the possible number of strength\nvalues. What then should d and Nb be?\nA conservative lower bound on the estimated ratio of thalamic to basal ganglia output\nneurons is on the order of 5:1 (Figure 1). As the number of thalamic neurons in our ring\nmodel here is T = d · b, so this ratio implies d = 5. And with that distance between\nstrength distributions, that implies a strength resolution of at least Nb ≥ 11 for each\nthalamic neuron to receive input from at least four basal ganglia output neurons.\nReferences\nAhmadian, Y., Rubin, D. 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