Optimal Balance of the Striatal Medium Spiny Neuron Network

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 and only when MSN generated inhibitory post-synaptic potentials (IPSPs) are realistically sized. Here we suggest a reason for this. We investigate how MSN network generated population activity interacts with temporally varying cortical driving activity, as would occur in a behavioural task. We find that at unrealistically high connectivity a stable winners-take-all type regime is found where network activity separates into fixed stimulus dependent regularly firing and quiescent components. In this regime only a small number of population firing rate components interact with cortical stimulus variations. Around connectivity a transition to a more dynamically active regime occurs where all cells constantly switch between activity and quiescence. In this low connectivity regime, MSN population components wander randomly and here too are independent of variations in cortical driving. Only in the transition regime do weak changes in cortical driving interact with many population components so that sequential cell assemblies are reproducibly activated for many hundreds of milliseconds after stimulus onset and peri-stimulus time histograms display strong stimulus and temporal specificity. We show that, remarkably, this activity is maximized at striatally realistic connectivities and IPSP sizes. Thus, we suggest the local MSN network has optimal characteristics – it is neither too stable to respond in a dynamically complex temporally extended way to cortical variations, nor is it too unstable to respond in a consistent repeatable way. Rather, it is optimized to generate stimulus dependent activity patterns for long periods after variations in cortical excitation.

shows cell raster plot time series segments and corresponding similarity matrices M (t 1 , t 2 ) from high and low connectivity simulations, clustered by the k-means algorithm applied to only one of the stimuli. In both simulations some cells show rates which vary depending on the stimulus however reproducible stimulus onset locked temporal patterning is absent at both high and low connectivity. At high connectivity spike processes appear Poissonian or more regular than Poissonian. At low connectivity activity is very bursty.

Effect of reduction in inhibitory neurotransmitter timescale τ g on IPSP
In Figure S2 we illustrate the affects of variation in the inhibitory neurotransmitter g(t) timescale τ g . Figure S2(a) contrasts the IPSPs generated in a postsynaptic cell close to firing threshold for connection strengths the same as in a connectivity ρ = 0.2 simulation with the synaptic strength parameter κ = 1. This means the peak synaptic conductance is 3.4/(0.2 × 50) = 0.34nS when τ g = 50 msec and 3.4/(0.2 × 20) = 0.86nS when τ g = 20. The time taken for the IPSP to decay to half its peak value is about 50 msec when τ g = 50 (red). Its peak value is about 250mV. When the timescale is reduced to τ g = 20 msec (green) the peak value is increased to about 400mV and the half life reduced to about 25 msec. However has explained in the main text the total quantity of neurotransmitter released by a presynaptic spike is the same in both cases independent of the timescale τ g .
In Figure S2(b) we show time series of neurotransmitter g(t) for a cell firing around 7 Hz for the two different values of neurotransmitter timescale τ g . As can be seen both time series have similar mean values g ≈ 0.007 as expected however the fluctuations around this value are much larger in the shorter timescale τ g = 20 case (green).

Time series examples for reduced rate model
In Figure S3 we show examples of time series segments from the deterministic reduced rate model. In these simulations the excitatory input is fixed for the duration of the simulation without stochastic fluctuations and without input switching. The parameters are the same as in the full network simulation investigated in Figure 6 of the main paper. The low connectivity ρ = 0.07 example Figure S3(b) shows chaotic temporal evolution. The high connectivity ρ = 0.75 example Figure S3(a) illustrates the rapid approach to a fixed point with cells firing at different rates. The connectivity ρ = 0.3 example Figure  S3(c) shows a periodic state.
Distribution of fixed points, periodic and chaotic states in reduced rate model In Figure S4 we show how the dynamical variance of cells' firing rate time series in 500 cell simulations of the reduced rate model depends on the network parameters connectivity ρ and connection strength κ. That is for each cell i we calculate where the expectation is taken over time t from t = 100 to t = 110 seconds in 1 msec steps. We then average σ 2 i across all cells i for each simulation. The bars indicate the spread in σ 2 i across cells i in each simulation. The simulations are further divided into those with positive Lyapunov exponents indicating unstable dynamics (black circles) and those with negative Lyapunov exponents indicating stable dynamics (red squares). These figures include the same simulations as Figure 7 of the main text. Figure S4 (a) shows the connectivity variation. The points at high connectivity ρ > 0.5 with zero variance (shown as variance 10 −10 ) and negative Lyapunov exponents (red squares) correspond to simulations where all cells firing rates find completely fixed levels. The points with negative Lyapunov exponents (red squares) and non-zero variances correspond to simulations where some cells have periodically varying firing rates and the rest fixed rates. The black circles correspond to chaotic states. In the transition connectivity regime 0.17 < ρ < 0.5 there are some periodic simulations, some fixed point simulations and some chaotic simulations. Below connectivity ρ < 0.17 there are very few fixed points and periodic points and most simulations are chaotic. Figure S4 (b) shows the connection strength variation. At low connection strength κ < 1 all simulations are fixed point. In the transition regime κ ≈ 1 there are a few periodic simulations. Above κ > 1 all simulations are chaotic.

Stimulus response remains stochastic in deterministic spiking network model
In the main text all simulations of the spiking network model were conducted with stochastic fluctuations in the excitatory driving. Here we demonstrate that even when there are no such fluctuations and the network simulation is entirely deterministic the network still responds in a stochastic way to stimulus presentation. Figure S5(a) shows a mean 8 second similarity matrix from a connectivity ρ = 0.16 deterministic 500 cell spiking network simulation under the 2 × 2 second input switching protocol. In this simulation stimulus A presented during periods t = 0 ∼ 2 and t = 4 ∼ 6 shows fairly strong stimulus onset locked reproducible dynamics for more than a second after stimulus onset. This is shown by the fact that similarity along the main diagonal in the following presentation of stimulus A, M (4 < t 1 < 6, 0 < t 2 < 2) remains quite sharply peaked until about t 1 = 5. Figure S5(b) shows the similarity matrix itself D(t 1 , t 2 ) (see Materials and Methods) for a 22 second segment from the time series. Stimulus A presentations occur at t = 4n ∼ 4n + 2, for n = 0, 1, 2, .... For example the block D(16 < t 1 < 18, 8 < t 2 < 10) shows the similarity between the 5 th presentation of stimulus A and the third presentation of stimulus A. As can be seen during some presentations of stimulus A the network response is strongly reproducible for long periods of up to two seconds after stimulus onset (for example D(8 < t 1 < 10, 0 < t 2 < 2)) while other presentations of stimulus A only show weak reproducibility (for example D(16 < t 1 < 18, 12 < t 2 < 14)). Thus even though the network simulation is entirely deterministic it still responds in a way which varies trial by trial. This could be an origin of error trials in behavioural tasks.