Cell Assembly Dynamics of Sparsely-Connected Inhibitory Networks: A Simple Model for the Collective Activity of Striatal Projection Neurons
[['The basal ganglia are critical brain structures for behavioral control , whose organization has been highly conserved during vertebrate evolution [1] .', 'Altered activity of the basal ganglia underlies a wide range of human neurological and psychiatric disorders , but the specific computations normally performed by these circuits remain elusive .', 'The largest component of the basal ganglia is the striatum , which appears to have a key role in adaptive decision-making based on reinforcement history [2] , and in behavioral timing on scales from tenths of seconds to tens of seconds [3] .', 'The great majority ( > 90% ) of striatal neurons are GABAergic medium spiny neurons ( MSNs ) , which project to other basal ganglia structures but also make local collateral connections within striatum [4] .', 'These local connections were proposed in early theories to achieve action selection through strong winner-take-all lateral inhibition [5 , 6] , but this idea fell out of favor once it became clear that MSN connections are actually sparse ( nearby connection probabilities ≃ 10–25% [7 , 8] ) , unidirectional and relatively weak [9 , 10] .', 'Nonetheless , striatal networks are intrinsically capable of generating sequential patterns of cell activation , even in brain slice preparations without time-varying external inputs [11 , 12] .', 'Following previous experimental evidence that collateral inhibition can help organize MSN firing [13] , an important recent set of modeling studies argued that the sparse connections between MSNs , though individually weak , can collectively mediate sequential switching between cell assemblies [14 , 15] .', 'It was further hypothesized that these connections may even be optimally configured for this purpose [16] .', 'This proposal is of high potential significance , since sequential dynamics may be central to the striatum’s functional role in the organization and timing of behavioral output [17 , 18] .', 'In their work [14–16] , Ponzi and Wickens used conductance-based model neurons ( with persistent Na+ and K+ currents [19] ) , in proximity to a bifurcation from a stable fixed point to a tonic firing regime .', 'We show here that networks based on simpler leaky integrate-and-fire ( LIF ) neurons can also exhibit sequences of cell assembly activation .', 'This simpler model , together with a novel measure of structured bursting , allows us to more clearly identify the critical parameters needed to observe dynamics resembling that of the striatal MSN network .', 'Among other results , we show that the duration of GABAergic post-synaptic currents is crucial for the network’s ability to discriminate different input patterns .', 'GABAergic currents are abnormally brief in mouse models of Huntington’s Disease ( HD ) [20] , and we demonstrate how this may produce the altered neural activity dynamics reported for symptomatic HD mice [21] .', 'Finally , dopamine loss weakens MSN-MSN interactions [22 , 23] , and our results help illuminate the origins of aberrant synchronization patterns in Parkinson’s Disease ( PD ) .'], ['The model is composed of N leaky integrate-and-fire ( LIF ) inhibitory neurons [24 , 25] , with each neuron receiving inputs from a randomly selected 5% of the other neurons ( i . e . a directed Erdös-Renyi graph with constant in-degree K = pN , where p = 5% ) [26] .', 'The inhibitory post-synaptic potentials ( PSPs ) are modeled as α-functions characterized by a decay time τα and a peak amplitude APSP .', 'In addition , each neuron i is subject to an excitatory input current mimicking the cortical and thalamic inputs received by the striatal network .', 'In order to obtain firing periods of any duration the excitatory drives are tuned to drive the neurons in proximity of the supercritical bifurcation between the quiescent and the firing state , similarly to [14] .', 'Furthermore , our model is integrated exactly between a spike emission and the successive one by rewriting the time evolution of the network as an event-driven map [27] ( for more details see Methods ) .', 'Since we will compare most of our findings with the results reported in a previous series of papers by Ponzi and Wickens ( PW ) [14–16] it is important to stress the similarities and differences between the two models .', 'The model employed by PW is a two dimensional conductance-based model with a potassium and a sodium channel [19] , our model is simply a current based LIF model .', 'The parameters of the PW model are chosen so that the cell is in proximity of a saddle-node on invariant circle ( SNIC ) bifurcation to guarantee a smooth decrease of the firing period when passing from the quiescent to the supra-threshold regime , without a sudden jump in the firing rate .', 'Similarly , in our simulations the parameters of the LIF model are chosen to be in proximity of the bifurcation from silent regime to tonic firing .', 'In the PW model the PSPs are assumed to be exponentially decaying , whereas we considered α-functions .', 'In particular , we are interested in determining model parameters for which uniformly distributed inputs I = {Ii} , where Ii ∈ [Vth , Vth + ΔV] , produce cell assembly-like sequential patterns in the network .', 'The main aspects of the desired activity can be summarized as follows:', '( i ) single neurons should exhibit large variability in firing rates ( CV > 1 ) ;', '( ii ) the dynamical evolution of neuronal assemblies should reveal strong correlation within an assembly and anti-correlation with neurons out of the assembly .', 'As suggested by many authors [9 , 28] the dynamics of MSNs cannot be explained in terms of a winners take all ( WTA ) mechanism , which would imply a small number of highly firing neurons , while the remaining would be silenced .', 'Therefore we searched for a regime in which a substantial fraction of neurons are actively involved in the network dynamics .', 'This represents a third criterion', '( iii ) to be fulfilled to obtain a striatum-like dynamical evolution .', 'Fig 1 shows an example of such dynamics for the LIF model , with three pertinent similarities to previously observed dynamics of real MSN networks [11] .', 'Firstly , cells are organized into correlated groups , and these groups are mutually anticorrelated ( as evident from the cross-correlation matrix of the firing rates reported in Fig 1', '( c ) ) .', 'Secondly , individual cells within groups show irregular firing as shown in Fig 1', '( a ) .', 'This aspect is reflected in a coefficient of variation ( CV ) of the inter-spike-intervals ( ISIs ) definitely greater than one ( as we will examine in following subsections ) , consistent with experimental observations for the dynamics of rat striatum in-vitro[7 , 9] .', 'Thirdly , the raster plot reported in Fig 1', '( b ) reveals that a large fraction of neurons ( namely , ≃ 93% ) is active .', 'The properties', '( i ) ,', '( ii ) , and', '( iii ) , characterizing MSN network activity , can be quantified in terms of a single scalar metric Q0 , as follows:', 'Q 0 ≡ 〈 C V 〉 N × σ ( C ( ν i , ν j ) ) × n * ; ( 1 )', 'where 〈⋅〉N denotes average over the ensemble of N neurons , n* = Nact/N is the fraction of active neurons Nact out of the total number , C ( νi , νj ) is the N × N zero-lag cross-correlation matrix between all the pairs of single neuron firing rates ( νi , νj ) , and σ ( ⋅ ) is the standard deviation of this matrix ( for details see Methods ) .', 'We expected that suitable parameter values for our model could be selected by maximizing Q0 .', 'Our metric is inspired by a metric introduced to identify the level of cluster synchronization and organization for a more detailed striatal microcircuit model [29] .', 'However , that metric is based on the similarity among the point-process spike trains emitted by the various neurons , whereas Q0 uses correlations between firing rate time-courses .', 'Furthermore , Q0 takes also in account the variability of the firing rates , by including the average CV in Eq ( 1 ) , an aspect of the MSN dynamics omitted by the metric employed in [29] .', 'Within biologically meaningful ranges , we find values of the parameters controlling lateral inhibition ( namely , the synaptic strength g and the the post-synaptic potential duration τα ) that maximize Q0 .', 'As we show below , the chosen parameters not only produce MSN-like network dynamics but also optimize the network’s computational capabilities , in the sense of producing a consistent , temporally-structured response to a given pattern of inputs while discriminating between very similar inputs .', 'In this sub-section we examine how network dynamics are affected by the strength of inhibitory connections ( Fig 2 ) .', 'When these lateral connections are very weak ( parameter g close to zero ) , the dominant input to each neuron is the constant excitation .', 'As a result , most individual neurons are active ( fraction of active neurons , n* , is close to 1 ) and firing regularly ( CV close to zero ) .', 'As lateral inhibition is made stronger , some neurons begin to slow down or even stop firing , and n* declines towards a minimum fraction of ≃ 50% ( at g = gmin ) , as shown in the lower panel in Fig 2', '( a ) .', 'As noted by Ponzi and Wickens [16] , this is due to a WTA mechanism: faster-firing neurons depress or silence the neurons to which they are connected .', 'This is evident from the distribution P ( I S I ¯ ) of the average interspike intervals ( I S I ¯ ) , which is peaked at low firing periods , and from the distribution of the CV exhibiting a single large peak at CV ≃ 0 ( as shown in the insets of S2a , S2b , S2d and S2e Fig ) .', 'As soon as g approaches gmin , the neuronal activity is no longer due only to the winners , but also the losers begin to play a role .', 'The winners are characterized by an effective input Wi which is on average supra-threshold , while their firing activity is driven by the mean current: winners are mean-driven[30] .', 'On the other hand , losers are on average below-threshold , and their firing is due to current fluctuations: losers are fluctuation-driven[30] .', 'For more details see S2c and S2f Fig . This is reflected in the corresponding distribution P ( I S I ¯ ) ( Fig 2', '( b ) , red curve ) .', 'The winners have very short I S I ¯s ( i . e . high firing rates ) , while the losers are responsible for the long tail of the distribution extending up to I S I ¯ ≃ 10 3 s .', 'In the distribution of the coefficients of variation ( Fig 2', '( b ) inset , red curve ) the winners generate a peak of very low CV ( i . e . highly-regular firing ) , suggesting that they are not strongly influenced by the other neurons in the network and therefore have an effective input on average supra-threshold .', 'By contrast the losers are associated with a smaller peak at CV ≃ 1 , confirming that their firing is due to large fluctuations in the currents .', 'Counterintuitively however , further increases in lateral inhibition strength result in increased neuronal participation , with n* progressively returning towards ≃ 1 .', 'The same effect was previously reported by Ponzi and Wickens [16] for a different , more complex , model .', 'When the number of active neurons returns almost to 100% , i . e . for sufficiently large coupling g > gmin , most of the neurons appear to be below threshold , as revealed by the distribution of the effective inputs Wi reported in S2c and S2f Fig . Therefore in this case the network firing is essentially fluctuation-driven , and the P ( I S I ¯ ) distribution is now characterized by a broader distribution and by the absence of the peak at short I S I ¯ ( as shown in Fig 2', '( b ) , blue line; see also S2a and S2d Fig ) .', 'Furthermore the single neuron dynamics is definitely bursting , as shown by the CV distribution now centered around CV ≃ 2 ( inset of Fig 2', '( b ) , blue line; see also S2b and S2e Fig ) .', 'The transition between these two dynamical regimes , occurring at g = gmin , is due to a passage from a state where some winner neurons were mean-driven and able to depress all the other neurons , to a state at g > >gmin where almost all neurons are fluctuation-driven and contribute to the network activity .', 'The transition occurs because at g < gmin the fluctuations in the effective input currents Wi are small and insufficient to drive the losers towards the firing threshold ( as shown in the insets of S2c and S2f Fig ) .', 'At g ≃ gmin the amplitude of the fluctuations becomes sufficient for some losers to cross the firing threshold and contribute to the number of active neurons .', 'This will also reduce the winners’ activity .', 'For g > >gmin the fluctuations of Wi are sufficient to lead almost all losers to fire and no clear distinction between losers and winners remains .', 'The reported results explain why the variability σ ( C ) of the cross-correlation matrix has a non monotonic behaviour with g ( as shown in the middle panel in Fig 2', '( a ) ) .', 'At low coupling σ ( C ) is almost zero , since the single neuron dynamics are essentially independent one from another , while the increase of the coupling leads to an abrupt rise of σ ( C ) .', 'This growth is clearly associated with the inhibitory action which splits the neurons into correlated and anti-correlated groups .', 'The variability of the cross-correlation matrix achieves a maximum value for coupling slightly larger than gmin , where fluctuations in the effective currents begin to affect the network dynamics .', 'At larger coupling σ ( C ) begins to decay towards a finite non zero value .', 'These results confirm that the most interesting region to examine is the one with coupling g > gmin , as already suggested in [16] .', 'The observed behaviour of CV , n* and σ ( C ) suggests that we should expect a maximum in Q0 at some intermediate coupling g > gmin , as indeed we have for both studied cases as shown in Fig 2', '( c ) and 2', '( d ) .', 'The initial increase in Q0 is due to the increase in CV and n* , while the final decrease , following the occurrence of the maximum , is essentially driven by the decrease of σ ( C ) .', 'For larger ΔV the neurons tend to fire regularly across a wider range of small g values ( see Fig 2', '( d ) ) , indicating that due to their higher firing rates a larger synaptic inhibition is required to influence their dynamics .', 'On the other hand , their bursting activity observable at large g is more irregular ( see the upper panel in Fig 2', '( a ) ; dashed line and empty symbols ) .', 'To assess whether parameters that maximize Q0 also allow discrimination between different inputs , we alternated the network back and forth between two distinct input patterns , each presented for a period Tsw .', 'During this stimulation protocol , we evaluated the state transition matrix ( STM ) and the associated quantity ΔMd .', 'The STM measures the similarity among the firing rates of the neurons in the network taken at two different times .', 'The metric ΔMd , based on the STM , has been introduced in [16] to quantify the ability of the network to distinguish between two inputs .', 'In particular , ΔMd is the difference between the average values of the STM elements associated with the presentation of each of the two stimuli ( a detailed description of the procedure is reported in the sub-section Discriminative and computation capability and in Methods ) .', 'To verify whether the ability of the network to distinguish different stimuli is directly related to the presence of dynamically correlated and anti-correlated cell assemblies , we generated a new metric , Qd .', 'This metric is defined in the same way as Q0 , except that in Eq ( 1 ) the standard deviation of the correlation matrix is replaced by ΔMd .', 'As it can be appreciated from Fig 2', '( c ) and 2', '( d ) the metrics Qd and Q0 behave similarly , indicating that indeed Q0 becomes maximal in the parameter range in which the network is most effectively distinguishing different stimuli .', 'We speculate that the emergence of correlated and anti-correlated assemblies contributes to this discriminative ability .', 'We note that we observed maximal values of Q0 for realistic lateral inhibition strengths , as measured from the post-synaptic amplitudes APSP .', 'Specifically , Q0 reaches the maximum at g = 4 ( g = 8 ) for ΔV = 1 mV ( ΔV = 5 mV ) corresponding to APSP = 0 . 368 mV ( APSP = 0 . 736 mV ) , comparable to previously reported experimental results [7 , 9 , 28] ( for more details see Methods ) .', 'In brain slice experiments IPSCs/IPSPs between MSNs last 5–20 ms and are mainly mediated by the GABAa-receptor [7 , 31] .', 'In this sub-section , we will examine the effect of the the post-synaptic time constant τα .', 'As τα is increased from 2 to 50 ms , the values of both metrics Q0 and Qd progressively increase ( Fig 3', '( a ) ) , with the largest variation having already occurred by τα = 20 ms . To gain more insights on the role of the PSP in shaping the structured dynamical regime , we show for the same network the distribution of the single cell CV , for τα = {2 , 9 , 20} ms ( Fig 3', '( b ) ) .', 'Narrow pulses ( τα ≃ 2 ms ) are associated with a distribution of CV values ranging from 0 . 5 to 1 , with a predominant peak at one .', 'By increasing τα one observes that the CV distributions shift to larger and larger CV values .', 'Therefore , one can conclude that at small τα the activity is mainly Poissonian , while increasing the duration of the PSPs leads to bursting behaviours , as in experimental measurements of the MSN activity [21] .', 'In particular in [21] , the authors showed that bursting activity of MSNs with a distribution P ( CV ) centered around CV ≃ 2 is typical of awake wild-type mice .', 'To confirm this analysis we estimated the distribution of CV2: A CV2 distribution with a peak around zero denotes very regular firing , while a peak around one indicates the presence of long silent periods followed by rapid firing events ( i . e . a bursting activity ) .', 'Finally a flat distribution denotes Poissonian distributed spiking .', 'It is clear from Fig 3', '( c ) that by increasing τα from 2 to 20 ms this leads the system from an almost Poissonian behaviour to bursting , where almost regular firing inside the burst ( intra-burst ) is followed by a long quiescent period ( inter-burst ) before starting again .', 'The distinct natures of the distributions of CV for short and long pulses raises the question of what mechanism underlies such differences .', 'To answer this question we analyzed the distribution of the ISI of a single cell in the network for two cases: in a cell assembly bursting regime ( corresponding to τα = 20 ms ) and for Poissonian unstructured behavior ( corresponding to τα = 2 ms ) .', 'We expect that even the single neurons should have completely different dynamics in these two regimes , since the distributions P ( CV ) at τα = 2 ms and 20 ms are essentially not overlapping , as shown in Fig 3', '( b ) .', 'In order to focus the analysis on the effects due to the synaptic inhibition , we have chosen , in both cases , neurons receiving exactly the same external excitatory drive Is .', 'Therefore , in absence of any synapses , these two neurons will fire with the same period ISI0 = τm log[ ( Is−Vr ) / ( Is−Vth ) ] = 12 ms , corresponding to a firing rate of 8 . 33 Hz not far from the average firing rate of the networks ( namely , 〈ν〉N ≃ 7–8 Hz ) .', 'Thus these neurons can be considered as displaying typical activity in both regimes .', 'As expected , the dynamics of the two neurons is quite different , as evident from the P ( ISI ) presented in Fig 4', '( a ) and 4', '( b ) .', 'In both cases one observes a long tailed exponential decay ( with decay rate νD ) of P ( ISI ) corresponding to a Poissonian like behaviour .', 'However the decay rate νD is slower for τα = 20 ms with respect to τα = 2 ms , namely νD ≃ 2 . 74 Hz versus νD ≃ 20 . 67 Hz .', 'Interestingly , the main macroscopic differences between the two distributions arises at short time intervals .', 'For τα = 2 ms , ( see Fig 4', '( b ) ) an isolated and extremely narrow peak appears at ISI0 .', 'This first peak corresponds to the supra-threshold tonic-firing of the isolated neuron , as reported above .', 'After this first peak , a gap is clearly visible in the P ( ISI ) followed by an exponential tail .', 'The origin of the gap resides in the fact that ISI0 > >τα , because if the neuron is firing tonically with its period ISI0 and receives a single PSP , the membrane potential has time to decay almost to the reset value Vr before the next spike emission .', 'Thus a single PSP will delay the next firing event by a fixed amount corresponding to the gap in Fig 4', '( b ) .', 'Indeed one can estimate analytically this delay due to the arrival of a single α-pulse , in the present case this gives ISI1 = 15 . 45 ms , in very good agreement with the results in Fig 4', '( b ) .', 'No further gaps are discernible in the distribution , because it is highly improbable that the neuron will receive two ( or more ) PSPs exactly at the same moment at reset , as required to observe further gaps .', 'The reception of more PSPs during the ramp up phase will give rise to the exponential tail in the P ( ISI ) .', 'In this case the contribution to the CV comes essentially from this exponential tail , while the isolated peak at ISI0 has a negligible contribution .', 'On the other hand , if τα > ISI0 , as in the case reported in Fig 4', '( a ) , P ( ISI ) does not show anymore a gap , but instead a continuous distribution of values .', 'This because now the inhibitory effects of the received PSPs sum up , leading to a continuous range of delayed firing times of the neuron .', 'The presence of this peak of finite width at short ISI in the P ( ISI ) plus the exponentially decaying tail are at the origin of the observed CV > 1 .', 'In Fig 4', '( e ) and 4', '( f ) the distributions of the coefficient C V 2 ( i ) are also displayed for the considered neurons as black lines with symbols .', 'These distributions clearly confirm that the dynamics are bursting for the longer synaptic time scale and essentially Poissonian for the shorter one .', 'We would like to understand whether it is possible to reproduce similar distributions of the ISIs by considering an isolated cell receiving Poissonian distributed inhibitory inputs .', 'In order to verify this , we simulate a single cell receiving K uncorrelated spike trains at a rate 〈ν〉N , or equivalently , a single Poissonian spike train with rate K〈ν〉N .', 'Here , 〈ν〉N is the average firing rate of a single neuron in the original network .', 'The corresponding P ( ISI ) are plotted in Fig 4', '( c ) and 4', '( d ) , for τα = 20 ms and 2 ms , respectively .', 'There is a remarkable similarity between the reconstructed ISI distributions and the real ones ( shown in Fig 4', '( a ) and 4', '( b ) ) , in particular at short ISIs .', 'Also the distributions of the C V 2 ( i ) for the reconstructed dynamics are similar to the original ones , as shown in Fig 4', '( e ) and 4', '( f ) .', 'Altogether , these results demonstrate that the bursting activity of coupled inhibitory cells is not a consequence of complex correlations among the incoming spike trains , but rather a characteristic related to intrinsic properties of the single neuron: namely , its tonic firing period , the synaptic strength , and the post-synaptic time decay .', 'The fundamental role played by long synaptic time in inducing bursting activity has been reported also in a study of a single LIF neuron below threshold subject to Poissonian trains of exponentially decaying PSPs [32] .', 'Obviously this analysis cannot explain collective effects , like the non trivial dependence of the number of active cells n* on the synaptic strength , discussed in the previous sub-section , or the emergence of correlations and anti-correlations among neural assemblies ( measured by σ ( C ) ) due to the covarying of the firing rates in the network , as seen in striatal slices and shown in Fig 1', '( c ) for our model .', 'To better investigate the influence of τα on the collective properties of the network we report in S4a and S4b Fig the averaged CV , σ ( C ) , n* and ΔMd for τα ∈ [2 , 50] ms . As already noticed , the network performs better in mimicking the MSN dynamics and in discriminating between different inputs at larger τα ( e . g . at 20 ms ) .', 'However , what is the minimal value of τα for which the network still reveals cell assembly dynamics and discriminative capabilities ?', 'From the data shown in S4a Fig one can observe that σ ( C ) and ΔMd attain their maximal values in the range 10 ms ≤ τα ≤ 20 ms . This indicates that clear cell assembly dynamics with associated good input pattern discrimination can be observed in this range .', 'However , the bursting activity is not particularly pronounced at τα = 10 ms , where 〈CV〉N ≃ 1 .', 'Therefore only the choice τα = 20 ms fulfills all the requirements .', 'Interestingly , genetic mouse models of HD revealed that spontaneuous IPSCs in MSNs has a reduced decay time and half-amplitude duration compared to wild-types [20] .', 'Our analysis clearly indicate that a reduction of τα results in more stochastic single-neuron dynamics , as indicated by 〈CV〉N ≃ 1 , as well as in a less pronounced structured assembly dynamics ( S4a Fig ) .', 'This resembles what observed for the striatum dynamics of freely behaving mice with symptomatic HD [21] .', 'In particular , the authors have shown in [21] that at the single unit level HD mice reveals a CV ≃ 1 in contrast to CV ≃ 2 for wild-type mice , furthermore the level of correlation among the neural firings was definitely reduced in HD mice .', 'A question that we have not addressed so far is: how do cell assemblies arise ?', 'Since the network is purely inhibitory it is reasonable to guess that correlation ( anti-correlation ) among groups of neurons will be related to the absence ( presence ) of synaptic connections between the considered groups .', 'In order to analyze the link between the correlation and the network connectivity we compare the clustered cross-correlation matrix of the firing rates C ( νi , νj ) ( shown in Fig 5', '( a ) ) with the associated connectivity matrix C i j ( reported in Fig 5', '( b ) ) .', 'The cross-correlation matrix is organized in k = 15 clusters via the k-means algorithm , therefore we obtain a matrix organized in a k × k block structure , where each block ( m , l ) contains all the cross-correlation values of the elements in cluster m with the elements in cluster', 'l . The connectivity matrix is arranged in exactly the same way , however it should be noticed that while C ( νi , νj ) is symmetric , the matrix C i j is not symmetric due to the unidirectional nature of the synaptic connections .', 'From a visual comparison of the two figures it is clear that the most correlated blocks are along the diagonal and that the number of connections present in these diagonal blocks is definitely low , with respect to the expected value from the whole matrix .', 'An exception is represented by the largest diagonal block which reveals , however , an almost zero level of correlation among its members .', 'We have highlighted in blue some blocks with high level of anti-correlations among the elements , the same blocks in the connectivity matrix reveal a high number of links .', 'A similar analysis , leading to the same conclusions was previously reported in [14] .', 'To make this comparison more quantitative , we have estimated for each block the average cross-correlation , denoted as 〈C〉ml , and the average probability pml of unidirectional connections from the cluster l to the cluster', 'm . These quantities are shown in Fig 5', '( c ) for all possible blocks .', 'The correlation 〈C〉ml decreases with the probability pml , and a linear fit to the data is reported in the figure as a solid black line .', 'However , there are blocks that are outliers with respect to this fit .', 'The blocks along the main diagonal ( black squares ) all have high correlation values 〈C〉mm and low probabilities pmm , smaller than the average probability p = 0 . 05 , shown as a dashed vertical red line in Fig 5', '( c ) .', 'An exception is represented by a single black square located exactly on the linear fit in proximity of p = 0 . 05 this is the large block with almost zero level of correlation among its elements previously identified .', 'Furthermore , the blocks with higher anticorrelation , denoted as blue triangles in the figure , have probabilities pml definitely larger than 5% .', 'The exceptions are two triangles lying exactly on the vertical dashed line corresponding to 5% .', 'This is due to the fact that the pml are not symmetric , and it is sufficient to have a large probability of connections in only one of the two possible directions between blocks m and l to observe anti-correlated activities between the two assemblies .', 'Having clarified that the origin of the assemblies identified from the correlations of the firing rates is directly related to structural properties of the networks , we would like to understand if the neurons belonging to the assemblies also share other properties .', 'In particular , we can measure the similarity of the neurons within each block ( m , l ) by estimating the block averaged similarity metrics eml introduced in Eq ( 14 ) in Methods .', 'This quantity measures how similar are the effective synaptic inputs Wi of two neurons in the network .', 'In Fig 5', '( d ) we report 〈C〉ml versus eml for all the blocks .', 'It is evident that the diagonal blocks ( black squares in the figure ) have a higher similarity value emm with respect to the average ( the vertical red dashed line in Fig 5', '( d ) ) .', 'Thus suggesting that correlated assemblies are formed by neurons with similar effective inputs and with few structural connections among them .', 'However , the observed anticorrelations are only due to structural effects since the most anticorrelated blocks ( blue triangles in Fig 5', '( d ) ) do not reveal any peculiar similarity value eml .', 'Similar results have been obtained by considering the neural excitability Ii instead of the effective synaptic input Wi .', 'In this sub-section we examine the ability of the network to perform different tasks: namely , to respond in a reproducible manner to stimuli and to discriminate between similar inputs via distinct dynamical evolution .', 'For this analysis we have always compared the responses of the network obtained for a set of parameters corresponding to the maximum Q0 value shown in Fig 2', '( d ) , where τα = 20 ms , and for the same parameters but with a shorter PSP decay time , namely τα = 2 ms . To check for the capability of the network to respond to cortical inputs with a reproducible sequence of states , we perform a simple experiment , following the protocol described in [15 , 16] , where two different inputs I ( 1 ) and I ( 2 ) are presented sequentially to the system .', 'Each input persists for a time duration Tsw and then the stimulus is switched to the other one and this process is repeated for the whole simulation time .', 'The raster plot measured during such an experiment is shown in Fig 6', '( a ) for τα = 20 ms . Whenever one of the stimuli is presented , a specific sequence of activations can be observed .', 'Furthermore , the sequence of emerging activity patterns is reproducible when the same stimulus is again presented to the system , as can be appreciated by observing the patterns encircled with black lines in Fig 6', '( a ) .', 'Furthermore , we can quantitatively compare the firing activity in the network at different times by estimating the STM .', 'Similarity is quantified by computing the normalized scalar product of the instantaneous firing rates of the N neurons measured at time ti and tj .', 'We observe that the similarity of the activity at a given time t0 and at successive times t0 + 2mTsw is high ( with values between 0 . 5 and 0 . 75 ) , while it is essentially uncorrelated with the response at times corresponding to the presentation of a different stimulus , i . e . at t0 + ( 2m−1 ) Tsw ( since the similarity is always smaller than 0 . 4 ) ( here , m = 1 , 2 , 3… ) .', 'This results in a STM with a periodic structure of period Tsw with alternating high correlated blocks followed by low correlated blocks ( see S5b Fig ) .', 'An averaged version of the STM calculated over a sequence of 5 presentations of I ( 1 ) and I ( 2 ) is shown in Fig 6', '( b ) ( for details of the cal
['Introduction', 'Results', 'Discussion', 'Methods']
['Striatal projection neurons form a sparsely-connected inhibitory network , and this arrangement may be essential for the appropriate temporal organization of behavior .', 'Here we show that a simplified , sparse inhibitory network of Leaky-Integrate-and-Fire neurons can reproduce some key features of striatal population activity , as observed in brain slices .', 'In particular we develop a new metric to determine the conditions under which sparse inhibitory networks form anti-correlated cell assemblies with time-varying activity of individual cells .', 'We find that under these conditions the network displays an input-specific sequence of cell assembly switching , that effectively discriminates similar inputs .', 'Our results support the proposal that GABAergic connections between striatal projection neurons allow stimulus-selective , temporally-extended sequential activation of cell assemblies .', 'Furthermore , we help to show how altered intrastriatal GABAergic signaling may produce aberrant network-level information processing in disorders such as Parkinson’s and Huntington’s diseases .']
['Neuronal networks that are loosely coupled by inhibitory connections can exhibit potentially useful properties .', 'These include the ability to produce slowly-changing activity patterns , that could be important for organizing actions and thoughts over time .', 'The striatum is a major brain structure that is critical for appropriately timing behavior to receive rewards .', 'Striatal projection neurons have loose inhibitory interconnections , and here we show that even a highly simplified model of this striatal network is capable of producing slowly-changing activity sequences .', 'We examine some key parameters important for producing these dynamics , and help explain how changes in striatal connectivity may contribute to serious human disorders including Parkinson’s and Huntington’s diseases .']
['action potentials', 'medicine and health sciences', 'neural networks', 'neurodegenerative diseases', 'membrane potential', 'genetic diseases', 'electrophysiology', 'neuroscience', 'animal models', 'model organisms', 'computational neuroscience', 'inhibitions', 'autosomal dominant diseases', 'huntington disease', 'research and analysis methods', 'computer and information sciences', 'animal cells', 'behavior', 'mouse models', 'clinical genetics', 'raster plots', 'cell biology', 'data visualization', 'physiology', 'neurons', 'single neuron function', 'neurology', 'biology and life sciences', 'cellular types', 'computational biology', 'neurophysiology']