{"title": "Cell Assemblies in Large Sparse Inhibitory Networks of Biologically Realistic Spiking Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 1273, "page_last": 1280, "abstract": "Cell assemblies exhibiting episodes of recurrent coherent activity have been observed in several brain regions including the striatum and hippocampus CA3. Here we address the question of how coherent dynamically switching assemblies appear in large networks of biologically realistic spiking neurons interacting deterministically. We show by numerical simulations of large asymmetric inhibitory networks with fixed external excitatory drive that if the network has intermediate to sparse connectivity, the individual cells are in the vicinity of a bifurcation between a quiescent and firing state and the network inhibition varies slowly on the spiking timescale, then cells form assemblies whose members show strong positive correlation, while members of different assemblies show strong negative correlation. We show that cells and assemblies switch between firing and quiescent states with time durations consistent with a power-law. Our results are in good qualitative agreement with the experimental studies. The deterministic dynamical behaviour is related to winner-less competition shown in small closed loop inhibitory networks with heteroclinic cycles connecting saddle-points.", "full_text": "Cell Assemblies in Large Sparse Inhibitory Networks\n\nof Biologically Realistic Spiking Neurons\n\nAdam Ponzi\n\nOIST, Uruma, Okinawa, Japan.\n\nadamp@oist.jp\n\nJeff Wickens\n\nOIST, Uruma, Okinawa, Japan.\n\nwickens@oist.jp\n\nAbstract\n\nCell assemblies exhibiting episodes of recurrent coherent activity have been\nobserved in several brain regions including the striatum[1] and hippocampus\nCA3[2]. Here we address the question of how coherent dynamically switching\nassemblies appear in large networks of biologically realistic spiking neurons in-\nteracting deterministically. We show by numerical simulations of large asymmet-\nric inhibitory networks with \ufb01xed external excitatory drive that if the network has\nintermediate to sparse connectivity, the individual cells are in the vicinity of a bi-\nfurcation between a quiescent and \ufb01ring state and the network inhibition varies\nslowly on the spiking timescale, then cells form assemblies whose members show\nstrong positive correlation, while members of different assemblies show strong\nnegative correlation. We show that cells and assemblies switch between \ufb01ring and\nquiescent states with time durations consistent with a power-law. Our results are in\ngood qualitative agreement with the experimental studies. The deterministic dy-\nnamical behaviour is related to winner-less competition[3], shown in small closed\nloop inhibitory networks with heteroclinic cycles connecting saddle-points.\n\n1 Introduction\n\nCell assemblies exhibiting episodes of recurrent coherent activity have been observed in several\nbrain regions including the striatum[1] and hippocampus CA3[2], but how such correlated activity\nemerges in neural microcircuits is not well understood. Here we address the question of how coher-\nent assemblies can emerge in large inhibitory neural networks and what this implies for the structure\nand function of one such network, the striatum.\nCarrillo-Reid et al.[1] performed calcium imaging of striatal neuronal populations and revealed spo-\nradic and asynchronous activity. They found that burst \ufb01ring neurons were widespread within the\n\ufb01eld of observation and that sets of neurons exhibited episodes of recurrent and synchronized burst-\ning. Furthermore dimensionality reduction of network dynamics revealed functional states de\ufb01ned\nby cell assemblies that alternated their activity and displayed spatiotemporal pattern generation.\nRecurrent synchronous activity traveled from one cell assembly to the other often returning to the\noriginal assembly; suggesting a robust structure. Assemblies were visited non-randomly in sequence\nand not all state transitions were allowed. Moreover the authors showed that while each cell assem-\nbly comprised different cells, a small set of neurons was shared by different assemblies. Although\nthe striatum is an inhibitory network composed of GABAergic projection neurons, similar types of\ncell assemblies have also been observed in excitatory networks such as the hippocampus. In a re-\nlated and similar study Sasaki et al.[2] analysed spontaneous CA3 network activity in hippocampal\nslice cultures using principal component analysis. They found discrete heterogeneous network states\nde\ufb01ned by active cell ensembles which were stable against external perturbations through synaptic\nactivity. Networks tended to remain in a single state for tens of seconds and then suddenly jump\nto a new state. Interestingly the authors tried to model the temporal pro\ufb01le of state transitions by a\n\n1\n\n\fhidden Markov model, but found that the transitions could not be simulated in this way. The authors\nsuggested that state dynamics is non-random and governed by local attractor-like dynamics.\nWe here address the important question of how such assemblies can appear deterministically in bio-\nlogically realistic cell networks. We focus our modeling on the inhibitory network of the striatum,\nhowever similar models can be proposed for networks such as CA3 if the cell assembly activity is\ncontrolled by the inhibitory CA3 interneurons. Network synchronization dynamics[4, 5] of random\nsparse inhibitory networks of CA3 interneurons has been addressed by Wang and Buzsaki[5]. They\ndetermined speci\ufb01c conditions for population synchronization including that the ratio between the\nsynaptic decay time constant and the oscillation period be suf\ufb01ciently large and that a critical mini-\nmal average number of synaptic contacts per cell, which was not sensitive to the network size, was\nrequired. Here we extend this work focusing on the formation of burst \ufb01ring cell assemblies\nThe striatum is composed of GABAergic projection neurons with fairly sparse asymmetric in-\nhibitory collaterals which seem quite randomly structured and that receive an excitatory cortical\nprojection[6]. Each striatal medium spiny neuron (MSN) is inhibited by about 500 other MSNs\nin the vicinity via these inhibitory collaterals and similarly each MSN inhibits about 500 MSNs.\nHowever only about 10%\u2212 30% of MSNs are actually excited by cortex at any particular time. This\nimplies that each MSN is actively inhibited by about 50\u2212150 cortically excited cells in general. It is\nimportant to understand why the striatum has this particular structure, which is incompatible with its\nputative winner-take-all role. We show by numerical computer simulation that very general random\nnetworks of biologically realistic neurons coupled with inhibitory Rall-type synapses[7] and indi-\nvidually driven by excitatory input can show switching assembly dynamics. We commonly observed\na switching bursting regime in networks with sparse to intermediate connectivity when the level of\nnetwork inhibition approximately balanced the external excitation so that the individual cells were\nnear a bifurcation point. In our simulations, cells and assemblies slowly and spontaneously switch\nbetween a depolarized \ufb01ring state and a more hyperpolarized quiescent state. The proportion of\nswitching cells varies with the network connectivity, peaking at low connection probability for \ufb01xed\ntotal inhibition. The sorted cross correlation matrix of the \ufb01ring rates time series for switching cells\nshows a fascinating multiscale clustered structure of cell assemblies similar to observations in[1, 2].\nThe origin of the deterministic switching dynamics in our model is related to the principle of winner-\nless competition (WLC) which has previously been observed by Rabinovich and coworkers[3] in\nsmall inhibitory networks with closed loops based on heteroclinic cycles connecting saddle points.\nRabinovich and coworkers[3] demonstrate that such networks can generate stimulus speci\ufb01c patterns\nby switching among small and dynamically changing neural ensembles with application to insect ol-\nfactory coding[8, 9], sequential decision making[10] and central pattern generation[11]. Networks\nproduce this switching mode of dynamical activity when lateral inhibitory connections are strongly\nnon-symmetric. WLC can represent information dynamically and is reproducible, robust against in-\ntrinsic noise and sensitive to changes in the sensory input. A closely related dynamical phenomenon\nis referred to as chaotic itinerancy,[12]. This is a state that switches between fully developed chaos\nand ordered behavior. The orbit remains in the vicinity of lower dimensional quasi-stable nearly pe-\nriodic \u201cattractor ruins\u201d for some time before eventually exiting to a state of high dimensional chaos.\nThis high-dimensional state is also quasi-stable, and after chaotic wandering the orbit is again at-\ntracted to one of the attractor ruins. Our study suggests attractor switching may be ubiquitous in\nbiologically realistic large sparse random inhibitory networks.\n\n2 Model\n\nThe network is composed of biologically realistic model neurons in the vicinity of a bifurcation\nfrom a stable \ufb01xed point to spiking limit cycle dynamical behaviour. To describe the cells we use\nthe IN a,p + Ik model described in Izhikevich[13] although any model near such a bifurcation would\nbe appropriate. The IN a,p + Ik cell model is two-dimensional and described by,\n\nC\n\ndVi\ndt\ndni\ndt\n\n= Ii(t) \u2212 gL(Vi \u2212 EL) \u2212 gN am\u221e(Vi)(Vi \u2212 EN a) \u2212 gkni(Vi \u2212 Ek)\n= (n\u221e \u2212 ni)/\u03c4n\n\n(1)\n\n(2)\n\nhaving leak current IL, persistent N a+ current IN a,p with instantaneous activation kinetic and a\nrelatively slower persistent K + current IK. Vi(t) is the membrane potential of the i \u2212 th cell, C\n\n2\n\n\fthe membrane capacitance, EL,N a,k are the channel reversal potentials and gL,N a,k are the maximal\nconductances. ni(t) is K + channel activation variable of the i \u2212 th cell. The steady state activation\ncurves m\u221e and n\u221e are both described by, x\u221e(V ) = 1/(1+exp{(V x\u221e\u2212 V )/kx\u221e}) where x denotes\nm or n and V x\u221e and kx\u221e are \ufb01xed parameters. \u03c4n is the \ufb01xed timescale of the K + activation variable.\nThe term Ii(t) is the input current to the i \u2212 th cell.\nThe parameters are chosen so that the cell is the vicinity of a saddle-node on invariant circle bifurca-\ntion. As the current Ii(t) in Eq.1 increases through the bifurcation point the stable node \ufb01xed point\nand the unstable saddle \ufb01xed point annihilate each other and a limit cycle having zero frequency is\nformed[13]. Increasing current further increases the frequency of the limit cycle. The input current\nIi(t) in Eq.1 is composed of both excitatory and inhibitory parts and given by,\n\nIi(t) = I c\n\n\u2212ksyn,ijgj(t)(Vi(t) \u2212 Vsyn).\n\n(3)\n\ni +(cid:88)\n\nj\n\nThe excitatory part is represented by I c\ni and models the effect of the cortico-striatal synapses. It has a\n\ufb01xed magnitude for the duration of a simulation, but varying across cells. In the simulations reported\ni are quenched random variables drawn uniformly randomly from the interval [Ibif , Ibif +\nhere the I c\n1] where Ibif = 4.51 is the current at the saddle-node bifurcation point. These values of excitatory\ninput current mean that all cells would be on limit cycles and \ufb01ring with low rates if the network\ninhibition were not present. In fact the inhibitory network may cause some cells to become quiescent\nby reducing the total input current to below the bifurcation point. Since the inhibitory current part\nis provided by the GABAergic collaterals of the striatal network it is dynamically variable. These\nsynapses are described by Rall-type synapses[7] in Eq.3 where the current into postsynaptic neuron\ni is summed over all inhibitory presynaptic neurons j and Vsyn and ksyn,ij are channel parameters.\ngj(t) is the quantity of postsynaptically bound neurotransmitter given by,\n\ndgj\ndt\n\n= \u0398(Vj(t) \u2212 Vth) \u2212 gj(t)\n\n\u03c4g\n\n(4)\nfor the j \u2212 th presynaptic cell. Here Vth is a threshold, and \u0398(x) is the Heaviside function. gj is es-\nsentially a low-pass \ufb01lter of presynaptic \ufb01ring. The timescale \u03c4g should be set relatively large so that\nthe postsynaptic conductance follows the exponentially decaying time average of many preceding\npresynaptic high frequency spikes.\nThe network structure is described by the parameters ksyn,ij = (ksyn/p)\u0001ijXij where \u0001ij is another\nuniform quenched random variable on [0.5, 1.5] independent in i and j. Xij = 1 if cells i and j\nare connected and zero otherwise. In the simulations reported here we use random networks where\ncells i and j are connected with probability p, and there are no self-connections, Xii = 0. ksyn is a\nparameter which is rescaled by the connection probability p so that average total inhibition on each\ncell is constant independent of p. All simulations were carried out with fourth order Runge-Kutta.\n\n3 Results\n\nFigure 1(a) shows a time series segment of membrane potentials Vi(t) for some randomly selected\ncells from an N = 100 cell network. The switching between \ufb01ring and quiescent states can clearly\nbe seen. Cells \ufb01re with different frequencies and become quiescent for variable periods before start-\ning to \ufb01re again apparently randomly. However the model has no stochastic variables and therefore\nthis switching is caused by deterministic chaos. As explained above the \ufb01ring rate is determined\nby the proximity of the limit cycle to the saddle-node bifurcation and can therefore be arbitrarily\nlow for this type of bifurcation. Since we have set the unit parameters so that all units are near\nthe bifurcation point even weak network inhibition is able to cause the cells to become quiescent\nat times. The parameter settings are biologically realistic[13] and MSN cells are known to show\nirregular quiescent and \ufb01ring states in vivo[14].\nThe complex bursting structure is easier to see from raster plots. A segment from a N = 100\ncell time series is shown in Fig.1(b). This \ufb01gure clearly shows attractor switching, or chaotic\nitinerancy[12], where a quasi-stable nearly-periodic state (an \u201cattractor ruin\u201d) is visited from higher\ndimensional chaos. To make this plot the cells have been ordered by the k-means algorithm with\n\ufb01ve clusters (see below). The cells are coloured according to the cluster assigned to them by the\nalgorithm. During the periodic window, most cells are silent however some cells \ufb01re continuously\n\n3\n\n\fFigure 1: (a) Membrane potential Vi(t) time series segment for a few cells from a N = 100 cell\nnetwork simulation with 20 connections per cell. Each cell time series is a different colour. (b)\nSpike raster plot from an N = 100 cell network simulation with 20 connections per cell. Each line\nis a different cell and the 71 cells which \ufb01re at least one spike during the period shown are plotted.\nCells are ordered by k-means with \ufb01ve clusters and coloured according to their assigned clusters.\n\nat \ufb01xed frequency and some cells \ufb01re in periodic bursts. In fact the cells which \ufb01re in bursts have\nbeen separated into two clusters, as can be seen in Fig.1(b), the blue and green clusters. These two\nclusters \ufb01re periodic bursts in anti-phase. Cell assemblies can also be seen in the chaotic regions.\nThe cells in the black cluster \ufb01re together in a burst around t = 17500 while the cells in the orange\ncluster \ufb01re a burst together around t = 16000. Fig.2(a) shows another example of a spike raster plot\nfrom a N = 100 cell network simulation where again the cells have been ordered by the k-means\nalgorithm with \ufb01ve clusters. Now cell assemblies, blue, orange and red coloured, can clearly be seen\nwhich appear to switch in alternation. This switching is further interrupted from time to time by the\ngreen and black assemblies.\nDue to the presence of attractor switching where cell assemblies can burst in antiphase we can\nexpect the appearance of strongly positively and strongly negatively correlated cell pairs. Correlation\nmatrices are constructed by dragging a moving window over a long spike time series and counting\nthe spikes to construct the associated \ufb01ring rate time series. The correlation matrix of the rate time\nseries is then sorted by the k-means method[2], which is equivalent to PCA. Each cell is assigned\nto one of a \ufb01xed number of clusters and the cells indices are reordered accordingly. Fig.2(b) shows\nthe cross-correlation matrix corresponding to the spike raster plot in Fig.2(a) with cells ordered the\nsame way. Within an assembly cells are positively correlated, while cells in different assemblies\noften show negative correlation.\nLarger networks with appropriate connectivites also show complex identity-temporal patterns. A\npatch-work of switching cell assembly clusters can be seen in the spike raster plot and corresponding\ncross-correlation matrix shown in Figs.2(c) and (d) respectively for a N = 500 cell system where the\ncells have been ordered by the k-means algorithm, now with 30 clusters. Any particular assembly\ncan seem to be burst \ufb01ring periodically for a spell before becoming quiescent for long spells. Other\ncell assemblies burst very occasionally for no apparent reason. Notice from the cross-correlation\nmatrices in Fig.2(b) and (d) that although some cell assemblies are positively correlated with each\nother, they have different relationships to other cell assemblies, and therefore cannot be combined\ninto a single larger assemblies.\nFig.2(c) reveals many cells switching between a \ufb01ring state and quiescent state. What is the struc-\nture of this switching state? To investigate this we analyse inter spike interval (ISI) distributions.\nShown in Fig.1(b) are three ISI distributions for three 500 cell network simulations in the sparse to\nintermediate regime with 30 connections per cell. The distributions are very broad and far from the\nexponential distribution one would expect from a Poisson process. They are consistent with a scale-\nfree power law behaviour for three orders of magnitude, but exponentially cut off at large ISIs due\nto \ufb01nite size effects. It is this distribution which produces the appearance of the complex identity-\ntemporal patterns shown in the 500 cell time series \ufb01gure in Fig.2(c) with the long ISIs interspersed\nwith the bursts of short ISIs. Power-law distributions are characteristic of systems showing chaotic\n\n4\n\n\fFigure 2: (a) Spike raster plot from all 69 cells in a 100 cell network with 20 connections per cell\nwhich \ufb01re at least one spike. The cells are ordered by k-means with \ufb01ve clusters and coloured\naccording to their assigned cluster. (b) Cross-correlation matrix corresponding to (a). The cells are\nordered by the k-means algorithm the same way as (a). Red colour means positive correlation, blue\nmeans negative correlation, colour intensity matches strength. White is weak or no correlation. (c)\nSpike raster plot from an N = 500 cell sparse network with 6 connections per cell. The 379 cells\nwhich \ufb01re at least one spike during the period shown are plotted. The cells are ordered by k-means\nwith 30 clusters. (d) Cross-correlation matrix corresponding to (c) with same conventions as (b).\n\nattractor switching and have been studied in connection with deterministic intermittency[15]. In-\ntermittency consists of laminar phases where the system orbits appear to be relatively regular, and\nbursts phases where the motion is quite violent and irregular. Interestingly a power-law distribution\nof state sojourn times was also observed in the hippocampal study of by Sasaki et al.[2] described\nabove. Plenz and Thiagarajan[16] discuss cortical cell assemblies in the framework of scale free\navalanches which are associated with intermittency[17].\nThe broad power-law distribution produces the temporal aspect of the complex identity-temporal\npatterns observed in the time series in Fig.2(c), however the fact that the cells show strong cross-\ncorrelation produces the spatial structure aspect. In the above we have shown how this structure can\nbe revealed using the k-means sorting algorithm. By combining the spikes of cells in a cluster into a\n\u201ccluster spike train\u201d preserving each spikes\u2019 timing we can study the ISIs of cluster spike time series.\nHowever the k-means algorithm produces a different clustering depending on the initial choice of\ncentroids. To control for this we perform the clustering many, here 200, times and combine the ISI\ntime series so generated into a single distribution. The black circles in Fig.3(a) show the cluster\nISI distribution after cells have been associated to clusters with the k-means algorithm with 10\nclusters. The cluster ISI distribution, like the individual cell ISI distribution, also shows a power-\nlaw over several orders of magnitude. This implies clusters also burst in a multiple scale way. The\nslope of the power law is greater than the individual cell result and the cut-off is lower as would\nbe expected when spike trains are combined. Nevertheless the distribution is still very broad. To\ndemonstrate this we perform a bootstrap type test where rather than making each cluster spike train\n\n5\n\n\fFigure 3: (a) Green, brown, blue: Three cell cumulative ISI distributions from 500 cell network sim-\nulations with 30 connections per cell, all cells combined. Log-log scale. The slope of the dashed line\nis \u22121.38. Black: ISI distribution for clusters formed by k-means algorithm corresponding to green\nsingle cell distribution. The slope of the solid line is \u22122.35. Red: ISI distribution for clusters formed\nfrom cells randomly corresponding to green single cell distribution. (b) Variation of connectivity for\n500 cell networks. Inset shows low connectivity detail. Each point calculated from a different net-\nwork simulation for observation period t = 2000 to t = 12000 msec. Red: Proportion of cells\nwhich \ufb01re at least one spike during the period. Blue: Proportion of cells \ufb01ring periodically. Black:\nAverage absolute cross-correlation (cid:104)|Cij|(cid:105) between all cells in network calculated from rate time\nseries constructed from counting spikes in moving window of size 2000 msec. Green: Coef\ufb01cient\nof variation (cid:104)CV (cid:105) of ISI distribution averaged across all cells in network rescaled by 1/3.\n\nfrom the cells associated to the cluster we perform the same k-means clustering to obtain correct\ncluster sizes but then scramble the cell indices, associating the cells to the clusters randomly. Again\nwe do this 200 times and combine all the results into one cluster ISI distribution. The red circles\nin Fig.3(a) show this random cluster ISI distribution. The distribution is much narrower than the\ndistribution obtained from the non-randomized k-means clustering. This demonstrates further that\nthe time series have a clustered structure which can be revealed by the k-means algorithm and that the\nclusters produced have a larger periods of quiescence between bursting than would be expected from\nrandomly associating cells, even when the cells themselves have power-law distributed ISIs. This\nbroadened distribution produced by the clustering re\ufb02ects the complex identity-temporal structure\nof the ordered spike time series \ufb01gures such as shown in Fig.2(c).\nThe model has several parameters, in particular the connection probability p. How does the forma-\ntion of switching assembly dynamics depend on the network connectivity? To study this we perform\nmany numerical simulations while varying p. As described above the synaptic ef\ufb01cacy is rescaled\nby the connection probability so the total inhibition on each cell is \ufb01xed and therefore effects arise\npurely from variations in connectivity.\nFig.3(b) (red) shows the proportion of cells which \ufb01re at least one spike versus average connections\nper cell for 500 cell network simulations. This quantity shows a transition around 5 connections per\ncell to state where almost all the network is burst \ufb01ring and then decays off to a plateau region at\nhigher connectivity. Fig.3(b) (blue) shows the proportion of cells \ufb01ring periodically. This is zero\nabove the transition. Below the transition a large proportion of cells are not inhibited and \ufb01ring\nperiodically due to the excitatory cortical drive, while another large proportion are not \ufb01ring at all,\ninhibited by the periodically \ufb01ring group. At high connectivities however most cells receive similar\ninhibition levels which leaves a certain proportion \ufb01ring. Fig.3(b) (green) shows the coef\ufb01cient of\nvariation CV of the single cell ISI distribution averaged across all cells and rescaled by 1/3. CV is\nde\ufb01ned to be the ISI standard deviation normalized by the mean ISI. It is unity for Poisson processes.\nBelow the transition CV is very low due to many periodic \ufb01ring cells. At high connectivities it is also\nlow and inspection of spike time series shows all cells \ufb01ring with fairly regular ISIs. In intermediate\nregions however this quantity can become very large re\ufb02ecting long periods of quiescent interrupted\nby high frequency bursting, as also re\ufb02ected in the single cell ISI distributions in Fig.3(a). Fig.3(b)\n(black) shows the average absolute cross-correlation (cid:104)|Cij|(cid:105) where Cij is the cross-correlation co-\n\n6\n\n\fef\ufb01cient between cells i and j \ufb01ring rate time series\u2019 and its absolute value is averaged across all\ncells. This quantity also shows the low connectivity transition but peaks around 200 connections per\ncell, where many cells are substantially cross-correlated (both positively and negatively). This is in\naccordance with the study of Wang and Buzsaki[5]. Fig.3(b) therefore displays an interesting regime\nbetween about 50 and 200 connections per cell where many cells are burst \ufb01ring with long periods\nof quiescence but have substantial cross-correlation. It is in this regime that spike time series often\nshow the complex identity-temporal patterns and switching cell assemblies exempli\ufb01ed in Fig.2(c).\n\n4 Discussion\n\nWe have shown that inhibitory networks of biologically realistic spiking neurons obeying deter-\nministic dynamical equations with sparse to intermediate connectivity can show bursting dynamics,\ncomplex identity-temporal patterns and form cell assemblies. The cells should be near a bifurca-\ntion point where even weak inhibition can cause them to become quiescent. The synapses should\nhave a slower timescale, \u03c4g > 10 in Eq.4, which produces a low pass \ufb01lter of presynaptic spiking.\nThis slow change in inhibition allows the bursting assembly dynamics since presynaptic cells do\nnot instantly inhibit postsynaptic cells, but inhibition builds up gradually, allowing the formation of\nassemblies which eventually becoming strong enough to quench the postsynaptic cell activity.\nAt low connectivities sets of cells with suf\ufb01ciently few and/or suf\ufb01ciently weak connections between\nthem will exist and these cells will \ufb01re together as an assembly due to the cortical excitation, if the\nrest of the network which inhibits them is suf\ufb01ciently quiescent for a period. Such a set of weakly\nconnected cells can be inhibited by another such set of weakly connected cells if each member of\nthe \ufb01rst set is inhibited by a suf\ufb01cient number of cells of the second set. When the second set ceases\n\ufb01ring the \ufb01rst set will start to \ufb01re. These assemblies can exist in asymmetric closed loops which\nslowly switch active set. Multiple \u201cfrustrated\u201d interlocking loops can exist where the slow switching\nof one loop will interfere with the dynamical switching of another loop; only when inhibition on one\nmember set is removed will the loop be able to continue slow switching, producing a type of neural\ncomputation. Furthermore any given cell can be a member of several such sets of weakly connected\ncells, as also described by Assisi and Bazhenov[18]. This can explain the \ufb01ndings of Carrillo-Reid\net.al.[1] who show some cells \ufb01ring with only one assembly and other cells \ufb01ring in multiple assem-\nblies. These cross-coupled switching assemblies with partially shared members produce complex\nmultiple timescale dynamics and identity-temporal patterning for appropriate connectivities.\nSwitching assemblies are most likely to be observed in networks of sparse to intermediate connectiv-\nities. This is consistent with WLC based attractor switching. Indeed networks with non-symmetric\ninhibitory connections which form closed circuits display WLC dynamics[3] and these will be likely\nto occur in networks with sparse to intermediate connectivities. The spike time series in Figs.2(a)\nand (c), indicate that cell assemblies switch non-randomly in sequence due to the deterministic at-\ntractor switching. This is in good agreement with Carrillo-Reid et al.[1] study of striatal dynamics\nand also with the Sasaki et al.[2] study of CA3 cell assemblies. Our time series and the cross-\ncorrelation matrices demonstrate that while most cells \ufb01re with only one particular assembly, some\ncells are shared between assemblies, as observed by Carrillo-Reid et al.[1]. We have shown that\ncells form assemblies of positively correlated cells and assemblies are negatively correlated with\neach other, in accordance with the similarity matrix results shown in Sasaki et al.[2].\nVery interestingly cell assemblies are predominantly found in a connectivity regime appropriate for\nthe striatum[6], where each cell is likely to be connected to about 100 cortically excited cells, sug-\ngesting the striatum may have adapted to be in this regime. Studies of spontaneous \ufb01ring in the\nstriatum also show very variable \ufb01ring patterns with long periods of quiescence[14], as shown in our\nsimulations at this connectivity. Based on studies of random striatal connectivity[6] we have simu-\nlated a random network without real spatial dimension. In support of this assumption Carrillo-Reid\net al.[1] \ufb01nd that correlated activity is spatially distributed, noting that neurons \ufb01ring synchronously\ncould be hundreds of microns apart intermingled with silent cells.\nAlthough we leave this point for future work the dynamics can also be affected by the details of\nthe spiking. Detailed inspection of the spike raster plot in Fig.1(b) con\ufb01rms three cells \ufb01ring with\nidentical frequency. Since these cells are driven by different levels of cortical excitation, the syn-\nchronization can only result from an entrainment produced by the spiking. This is possible in cells\nwith close \ufb01ring rates since the effect an inhibitory spike has on a post-synaptic cell depends on\n\n7\n\n\fthe post-synaptic membrane potential[13, 19]. In this way the spiking can affect cluster formation\ndynamics and may prolong the lifetime of visits to quasi-stable periodic states. The coupling of as-\nsembly dynamics and spiking may be relevant for coding in the insect olfactory lobe for example[9].\nThe striatum is the main input structure to the basal ganglia (BG). Correlated activity in cortico-basal\nganglia circuits is important in the encoding of movement, associative learning, sequence learning\nand procedural memory. Aldridge and Berridge[21] demonstrate that the striatum implements ac-\ntion syntax in rats grooming behaviour. BG may contain central pattern generators (CPGs) that\nactivate innate behavioral routines, procedural memories, and learned motor programs[20] and re-\ncurrent alternating bursting is characteristic of cell assemblies included in CPGs[20]. WLC has been\napplied to modeling CPGs[11]. Our modeling suggests that complex switching dynamics based in\nthe sparse striatal inhibitory network may allow the generation of cell assemblies which interface\nsensory driven cortical patterns to dynamical sequence generation. 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