^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: AP JRW. Performed the experiments: AP. Analyzed the data: AP. Wrote the paper: AP JRW.

Slowly varying activity in the striatum, the main Basal Ganglia input structure, is important for the learning and execution of movement sequences. Striatal medium spiny neurons (MSNs) form cell assemblies whose population firing rates vary coherently on slow behaviourally relevant timescales. It has been shown that such activity emerges in a model of a local MSN network but only at realistic connectivities of

The striatum forms the main input to the Basal Ganglia (BG), a subcortical structure involved in reinforcement learning and action selection. It is

The striatum forms the main input to the Basal Ganglia (BG), a subcortical structure involved in reinforcement learning and action selection. It is

Many studies of neural response to sensory stimuli and behavioural task events throughout the brain have found that cells display large highly repeatable variations in firing rate on slow behaviourally relevant time scales. In the striatum tonic and phasic MSN activity patterns have been observed locked to task

Since MSN network connectivity is sparse and weak it has been assumed in-vivo MSN firing patterns simply reflect cortical driving. Indeed if the roughly 10000 cortical inputs an MSN receives covary, even weakly

Indeed recent work seems to support the hypothesis that phasic in-vivo MSN activity can be partially generated internally within the striatum. Adler et al.

Several other recent studies have suggested the possibility that rather than acting independently MSNs may act coherently in cell assemblies

In recent modeling work

Here we investigate how sudden switches in cortical driving, as might occur in sensory driven behavioural tasks interacts with MSN network generated chaotic cell assembly activity. We show that stimulus specific cell assemblies can be reliably activated in sequence locked to stimulus switch times, resulting in slowly varying peri-stimulus time histograms (PSTH). Thus rather than generating a static stimulus dependent activity pattern we suggest the local MSN network is optimized to generate stimulus dependent dynamical activity patterns for long time periods after variations in cortical excitation. We investigate how this activity depends on network parameters and find that MSN task modulation is optimized in a marginally stable transition regime which occurs at striatally relevant connectivities and synaptic strengths. We discuss how these properties may be utilized in temporally delayed reinforcement learning tasks strongly recruiting the striatum.

In this section we illustrate stimulus onset locked cell assembly dynamics using an example time series. We show that the MSN network can generate prolonged sequences in response to sudden changes in otherwise constant cortical stimuli. Thus we show that the MSN network produces a dynamic sequence rather than a static state of active and quiescent cells due to the MSN network dynamics rather than the cortical drive.

In

Cell raster plot time series segment for the

In order to quantify the reproducibility of the dynamics we calculate the two-time firing rate

We denote by

The blocks

The blocks

These results demonstrate that an inhibitory spiking MSN network model can generate sequential patterns of activity for several hundred msecs after stimulus onset which are reproducible across different presentations of the same stimulus, but different for different stimuli. This is true even though the excitatory input strengths are fixed for the duration of a stimulus (except for random fluctuations). Thus the activation of cells is not simply determined by the input strengths. If this were the case (roughly speaking) the most strongly excited cells in any particular stimulus would remain active throughout that stimulus period while the least strongly activated would remain quiescent throughout the stimulus. Since the mean excitatory input strength is the same in both stimuli the onset locked patterns result only from the redistribution of excitation across MSNs; an increase in mean excitation level is not required. This is because cells are balanced close to firing threshold where even small variations in input drive cause a large change in the distribution and temporal evolution of activity across the inhibitory asymmetrically connected network. Thus balanced network activity provides cells with a large diversity of strong temporal responses to a given stimulus, rather than generating a static state of active and quiescent cells. Moreover clusters formed from many cells can also display this behaviour as observed in the time series

Recognition of stimuli through sequential activations remains stochastic however; on some trials a stimulus fails to generate its normal patterns. These failures may correspond to error trials in a behavioural task. Stochastic stimulus recognition is not due to the random fluctuations in excitation, but an effect of the chaotic network dynamics, also occurring in deterministic spiking network simulations as described in Supplemental

We have demonstrated that stimulus onset locked reproducible dynamics can occur in network simulations, but how does it depend on the network parameters such as connectivity and connection strength? To investigate these issues quantitatively we calculate mean similarity profiles for simulations of 500 cell networks. In previous work

(a) Mean similarity profiles

The late epoch

The behaviour is different at epochs

We now investigate how reproducibility of dynamical evolution depends on network connectivity. As explained in the Model section when we vary connectivity

The reproducibility at epoch

The quantity

(a) Strength of stimulus onset locked reproducible dynamics

In the Model section we explain that the time scale of inhibitory neurotransmitter decay is set by the parameter

We can also ask how the reproducibility of stimulus onset locked dynamics,

(a) Strength of stimulus onset locked reproducible dynamics

In

We have shown that the reproducibility of stimulus onset locked dynamical evolution and stimulus distinguishability are optimized in the striatally relevant parameter region of connectivity and connection strength. We now investigate why this should be. Here we show that the peaks occur near a transition in network activity which occurs in the striatally relevant parameter region and demonstrate the nature of the transition. In this section we investigate 500 cell network simulations under

The black points in

(a) Black circles: minimum observed ISI for each active cell in network simulations of different connectivity. Red line: mean of minimum observed ISI across all cells for each network simulation. Synaptic strength parameter

On the other hand at low connectivity the minimum ISI distribution does not show the quiescent component. The transition from a broad distribution to a narrow one appears to occur fairly suddenly around

The coefficient of variation (CV) of a cell's ISI distribution, defined as the cell's ISI standard deviation normalized by its mean ISI, also reveals the connectivity dependent transition.

The proportion of active cells (those that fire at least three spikes in the 168 second observation period) also demonstrates the connectivity dependent transition. This quantity (

Thus the network shows a fairly sharp transition from a regularly firing winners-take-all type regime where a proportion of cells are permanently quiescent to a regime where almost all cells are involved in bursty activity. Remarkably actual striatal connectivity of around

At very low

Remarkably again the transition between the winners-take-all like regime and the bursty active regime appears to be close to

Finally, as an illustration of the different activity in the two regimes,

(a,b) Firing rate time series segments based on 400 msec moving window for several randomly chosen cells from 500 cell network simulations under constant (randomly fluctuating) excitation without stimulus switching. Inhibitory neurotransmitter timescale

This observation can be quantified by the relative entropy

Above we have shown that the MSN network displays a transition between a bursty active regime and a winners-take-all like regime as connectivity and connection strength are varied. The transition occurs at striatally relevant parameter settings. Here we demonstrate that the rate dynamics generated by the MSN network model is unstable and chaotic in the bursty active regime but stable in the winners-take-all like regime and thus marginally stable at striatally relevant parameter settings.

The postsynaptically bound inhibitory neurotransmitters

The deterministic rate model shows a very similar qualitative dependence of the number of active cells on connectivity (

(a,b) Black circles : proportion of active cells. Red diamonds : mean relative entropy

The reduced model is deterministic and since it also lacks the strong instability of the spike generating mechanism we are able to compute the maximal Lyapunov exponent for the rate dynamics of 500 cell networks. This quantity characterises the stability of the rate dynamics. When it is positive the network rate dynamics is chaotic. When it is negative however the network has a found a fixed distribution of firing rates or alternatively some, or all, of the cells firing rates may be varying periodically. As can be seen by the blue crosses in

Time series examples from simulations of this reduced rate model displaying fixed point, periodic and chaotic activity are shown in Supplemental

Above we have demonstrated that temporally extended reproducible sequential dynamics can occur locked to stimulus switches. We have shown this activity occurs maximally near a transition in network activity where rate dynamics is marginally stable and which occurs in the striatally relevant parameter range. However in principle sequential activity could be mediated by a chain of single cells activated in sequence. Coherent activity of cell assemblies

The cell spike raster plot time series segment from the intermediate connectivity,

To investigate this further here instead of using k-means clustering we employ principal component analysis (PCA) of 500 cell network simulations. Principal components are linear combinations of single cell firing rates with fixed coefficients such that the resultant component activity time series are uncorrelated with each other. PCA is closely related

Using component analysis we can demonstrate that network dynamics can evolve in a much smaller dimensional space than the number of cells

(a,c,e) PSTH for several principal components (see key) locked to stimulus onset in the

At high connectivity,

These observations are reflected in the corresponding power spectral density (PSD) of the first ten components shown in

In contrast to the high connectivity situation at very low connectivity,

This can also be seen by the weakening of the

The situation is more interesting in the intermediate connectivity,

The increased complexity of the population dynamics is also apparent in the PSD,

The stimulus locking of population activity components described above can be quantified by the variance of the component PSTH fluctuations, here termed ‘PSTH variance’, and the variance of the component time series fluctuations around the mean PSTH activity, here termed ‘noise variance’ calculated across the first 400 msec after stimulus onset (see

In

(a) Component PSTH variance (dashed) and noise variance (solid) versus component number for three simulations of different connectivity

To quantify noise suppression in

In this paper we investigate how a minimalistic model of a local striatal MSN network responds to variations in cortical driving.

We first illustrate using a spike raster plot and mean similarity matrix that the MSN network model can display cell assembly population dynamics locked to stimulus onset times, as previously demonstrated in

We also investigate how stimulus distinguishability depends on IPSP size. Soon after stimulus onset the current stimulus is only weakly distinguishable from the previous one for all connection strengths. Distinguishably increases with time elapsed from stimulus onset. Most remarkably we find that the background activity (at long times after stimulus onset) generated by different stimuli shows a maximal distinguishability and this maximum occurs at striatally relevant IPSP size. In the striatally relevant parameter regime stimulus distinguishability takes about a second after stimulus onset to saturate at its maximal value.

To shed light on the origin of these optimal properties we investigate how the network generated dynamical activity of 500 cell network simulations under constant (fluctuating) excitatory drive, without input switching, depends on connectivity and connection strength. We find a transition in network generated dynamical activity around

To understand the network transition in more detail we investigate a simplified deterministic model of the network rate dynamics with parameters set exactly as in the full model. We are able to accurately reproduce the connectivity and connection strength dependence of network statistical quantities as well as the transition at striatally relevant parameter settings. We also numerically compute the maximal Lyapunov exponent and show that the network is marginally stable at striatally relevant parameter settings, separating a chaotic from a stable regime. In the stable regime the vast majority of network simulations show fixed point dynamics, especially at high connectivity, (see Supplemental

There are quantitative differences in the behaviour of the relative entropy and proportion of active cells between the rate and spiking models however. This is mainly due to the absence of dynamical effects induced by the spiking. Spiking causes noisy fluctuations around the fixed point states which reduces the relative entropy and may affect stability of attractors in the rate model. The periodic dynamical states are less likely to be observed in the full spiking network. Also transient periods of spike phase locking which may occur in the full spiking model

We next ask whether stimulus onset locked reproducible dynamics is mediated by single cells or by MSN cell assemblies with coherent slowly varying rates. To investigate this we apply principal component analysis to firing rate time series generated using a long 100 msec time window. Temporal variation in principal components is generated by the coherent activity of populations of cells. We show that at high connectivity only the first three population components show strong dependence on cortical variations. The first component represents the winning set of cells while the next two only activate transiently at stimulus switches. Network dynamics appears very stable and activated components rapidly relax between two fixed point states, one for each stimulus, characterised by different stimulus dependent distributions of regularly firing and quiescent cells across the network. As connectivity decreases more and more population components display reproducible dynamics after stimulus switches, peaking at around 10 at striatally relevant connectivity. The temporal variations of these components are generated by the coherent activation and deactivation of different subpopulations of cells which inhibit and disinhibit each other. At connectivities near the transition the network successively visits different transient distributions of active and quiescent cells before eventually finding a stable distribution. As connectivity decreases further population components appear to become unstable, wandering apparently randomly without locking to stimulus onset times. Thus cortical driving interacts maximally with network generated population activity at striatally relevant connectivity.

Now we discuss how these results can be explained within the framework of dynamical systems theory. There have been many investigations of dynamical regime transitions in networks of excitatory and inhibitory neurons. Regimes of synchronous and asynchronous irregular activity as well as oscillatory regimes have been found

In more recent work closely related to the present study, Buckley and Nowotny

Here we offer the following rough explanatory scenario. Our simulations of the deterministic reduced rate network suggest that the phenomenon we observe here is related to ‘critical slowing down’ occuring in marginally stable weakly chaotic transient dynamics close to the edge of chaos. Indeed (at least) two factors seem relevant for the generation of complex reproducible dynamics in the present random network model under the periodic forcing of the stimulus switching. First the dynamical trajectories generated by the network dynamics should remain quite complex and high dimensional for long periods after stimulus onset. If this is not the case multiple different states in a sequence cannot be discriminated or the elapsed time represented in this random network model. Second network dynamics in the periodically forced system should be stable with period of the forcing stimulus. The stability of dynamics under periodic forcing depends of course on the stability of dynamics generated by the autonomous network in both the stimuli in the absence of forcing. However it also depends on other factors such as the period of the forcing stimulus. In general periodic driving can cause stable activity states to become chaotic and vice-versa. Indeed Rajan et al.

One way in which activity in the transition regime between stable and unstable behaviour can be both complex and reproducible is due to temporally extended activity which would be transient to a stable fixed point in the unforced system. Indeed deep in the winners-take-all regime, far from the transition, network activity in the unforced system is characterised by a very stable stimulus dependent fixed point. In the periodically forced system after the excitatory input is switched the system moves to a new fixed point. The system moves rapidly between the fixed points due to their strong stability and with a highly reproducible trajectory due to the consistency of initial state across repeat stimulus presentations. Reproducibility is reflected in the strong noise suppression seen in all activated components (

On the other hand deep in the unstable regime reproducible stimulus locked dynamics does not occur even in the completely deterministic reduced rate network simulations (data not shown.) Here dynamical activity is complex and high dimensional, requiring many principal components to explain is variance, and thus can easily generate a sequence of strongly differing states. However since nearby trajectories rapidly diverge the network activity state at stimulus onset is strongly varying across repeat presentations and reproducibility is lost.

Transient activity in the unforced system may be complex and higher dimensional close to the transition due to the proximity of periodic and chaotic states and the prescence of attractor ruins. Attractor ruins are regions of phase space where attractors are weakly destabilized and close to which the flow is still very slow

This scenario is consistent with the observation that the transition seems to occur when the network is just balanced, as discussed above. In the winners-take-all state the permanent quiescent component allows the remaining active cells to fire fairly regularly, thus reducing the mean

Suggestions of critical dynamics can be seen in the the PSD of the higher components in the intermediate (

Indeed we observe (

Neural activity has often been modeled as a marginally stable critical process. Usually this is based on spiking activity. For example in a ‘critical branching process’

We have shown that in the vicinity of the transition the network displays optimal properties. A variety of optimal properties have been associated with marginally stable and critical behaviour in neural systems

Chaotic balanced networks

Many

In agreement with this work a variety of diverse response profiles with phasic activity peaks covering a wide spectrum of delays after task events has been observed in such tasks

Recently Adler et al.

The principal components representing population activities which we study here might be functionally and behaviourally relevant themselves and their activity might be indirectly observed through local field potential activity. Component activities are weighted summations of MSN activities with both positive and negative coefficients, a computation which could perhaps be performed by inhibition and disinhibition of globus palidus targets. In theory their activation could be utilised by the animal to represent stimulus onset and offset in behavioural tasks. Waves of cell assemblies could be used by an animal to mark time epochs from salient stimulus switches as well as the directionality of the switching and modulate central pattern generators by inhibition and disinhibition of downstream targets.

In the simulations reported here we use excitatory drive with a fairly broad distribution of excitations across cells, quantified by the input specificity parameter

Indeed experimental studies show that learning to perform procedural tasks alters neural firing patterns in the sensorimotor striatum as behaviour becomes more stereotyped

The model may be tested experimentally by comparing single cell MSN responses to manipulations of cortical input induced by stimulus changes in behavioural tasks, or by optogenetic activation in slices, before and after exclusively blocking lateral inhibition between MSNs. If blocking lateral inhibition changes MSN time courses then the model is supported. Due to noise suppression we also predict a sudden decrease in MSN Fano factors after stimulus onset which may be removed by blocking inhibition. However stimulus onset locked dynamics only occurs for stimuli of sufficient specificity

There are many good models of the striatum, for example

The network is composed of model MSNs with parameters set so they are in the vicinity of a bifurcation from a stable fixed point to spiking limit cycle dynamical behaviour

All the parameters are set as in

The input current

We describe the inhibitory MSN component

The MSN network synapses are described by Rall-type synapses

The representation of the MSN network is determined by the synaptic strengths

(i) Network structure. This is determined by

Knowledge of the mean size of an IPSP generated on a postsynaptic cell by a presynaptic cell together with the expected quantity of active inhibitory inputs gives us the expected level of total inhibition on a cell. In our network simulations we want to respect this level of total inhibition and thus use the approximately correct IPSP size (see below) and approximately the correct number of active synapses. In our random network simulations we can respect the figure of 450 connections by choosing a network of size

How do we make a striatally relevant choice for

Since MSN network structure within a striosome does not indicate anything other than a random process of connection growth we connect pairs of cells randomly with probability

(ii) However in this calculation there is a factor we have not yet included. This arises from the fact that during the course of any particular type of behaviour only a small proportion of MSNs are ever excited to levels above firing threshold by cortical or thalamic excitation. Such cells which can never fire can be left out of simulations altogether. Based on studies

(iii) Connection strength and IPSP size. Synaptic conductance and IPSP size are controlled by the factor

The factor

IPSPs generated by a spike on a postsynaptic MSN close to firing threshold tend to have peak sizes of around

(iv) IPSP time course. In our network model we intend to reproduce the time course of recovery of the membrane potential to firing threshold after a spike from another MSN. This, together with the peak IPSP size and quantity of cortically excited incoming synapses, controls the total inhibition a postsynaptic cell receives from presynaptic spiking. This is controlled by the dynamics of the inhibitory neurotransmitter

When changing the timescale

We model the excitatory driving

We assume the input spikes

MSN cells are each contacted by around 10000 cortical and thalamic cells and we therefore set

Here we investigate how the MSN network model responds to the simplest kind of temporally varying cortical input. This is just a sequence of different stimuli, such as might occur in a visual attentional task where a shape suddenly changes colour, for example. To model this we simply change all the cortical input rates

For each stimulus

We have chosen to use the Pareto distribution as a device to produce a large variation in excitation strength across MSN cells as only the simplest of several possibilities. There are others which may be biologically plausible, for example correlation in inputs to a single cell

In this paper we do not vary the parameters of the Pareto distribution

All simulations of the spiking network model were carried out with the stochastic weak second order Runge-Kutta integrator described in

Here we explain how the k-means algorithm is used in this paper. The number of clusters

An informative way to visualize at the time series is using firing rate similarity matrices,

Mean similarity profiles

Components are generated from the correlation matrix of firing rates of all

To calculate these quantities first calculate the

The relative entropy (Kullback-Liebler divergence) on two normalized distributions

The reduced rate model is obtained from the equation for the postsynaptically bound neurotransmitters

Statistical quantities are calculated using procedures analogous to those of the full spiking model. The proportion of active cells is calculated from the number of quiescent cells - those whose firing rate does not exceed a small value in the simulation period. A similar procedure is used to calculate the relative entropy for the reduced rate model (see above). The rates of all cells are sampled every time increment of the numerical integrator to generate the appropriate distributions. The minimum bin rate is set to zero and the maximum bin rate slightly greater than the observed maximum rate in the simulation. Quiescent cells, whose rates never exceed a small value, are not included in the relative entropy calculation. The maximal Lyapunov exponent is calculated in the standard way, as described in

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We would like to thank Rosario Mantegna for useful comments on a previous version of this work.