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Excitatory Population. External Input (Exc.). Inhibitory Population. III-28 [122] Spike Pattern Distributions in Model Cortical Networks Joanna Tyrcha, Stockholm University, Stockholm; John Hertz, Nordita, Stockholm/Copenhagen. Outline. Modeling the distribution of spike patterns.
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Excitatory Population External Input (Exc.) Inhibitory Population III-28 [122] Spike Pattern Distributions in Model Cortical NetworksJoanna Tyrcha, Stockholm University, Stockholm; John Hertz, Nordita, Stockholm/Copenhagen Outline Modeling the distribution of spike patterns A spin glass state? N-dependence is qualitatively consistent with spike patterns being sampled from a Sherrington-Kirkpatrick (SK) model: Cross-correlations between neurons measured in network simulations and experiments: -- for stationary balanced network 0.0052±0.0328 -- for highly stimulus-driven network 0.0086 ± 0.0278 -- in experiments ~0.01 [Schneidmann et al, Nature(2006)] Finding distribution of spike patterns with the observed cross-correlations: fit with SK spin glass model with E[Jij] > 0, std[Jij] falls off ~ no spin glass phase for our data Have sets of spike patterns {Si}k for spike/no spike(we use10-ms bins) (temporal order irrelevant) Construct a distribution P[S] that generates the observed patterns (i.e., has the same correlations) Simplest nontrivial model (Schneidman et al, Nature 440 1007 (2006), Tkačik et al, arXiv:q-bio.NC/0611072): Phase diagram of the SK model: axes: is the total “field” (external + internal) an infinite-range (mean field) model: Jij and hi are normally distributed with Get spike data from simulations of model network has normal and spin-glass phases; our data are in the normal phase order parameter equations: parametrized by Jij, hi; partition function Z normalizes distribution Inversion of Thouless-Anderson-Palmer equations: mean magnetization mean square magnetization 2 populations in network: Excitatory, Inhibitory Excitatory external drive HH-like neurons, conductance-based synapses Random connectivity:Probability of connection between any two neurons is c = K/N, where N is the size of the population and K is the average number of presynaptic neurons. Results here for c = 0.1, N = 1000 (Tanaka, PRE 58 2302 (1998)) TAP free energy: function of local “magnetizations” mi =E[Si]: => “TAP equations” Inversion procedure: Estimatemi and Cij from data: mi =<Si>,Cij = <SiSj> - mi mj Invert C matrix and solve for Jij in Solve TAP equations for hi: We do this for subsets of neurons of sizes N = 10-800 normal phase SG phase Correlation statistics for the SK model mean correlation matrix element: variance of correlation: where Solve for J0 and J1: Mean and variance of Jij’s are The inferred model is not in the spin glass phase for any N studied. Response to tonic input rapidly-varying stimulus response inhibitory (100) 16.1 Hz excitatory (400) 7.9 Hz inhibitory (100) 15.1 Hz excitatory (400) 8.6 Hz external population rate Perspective: Fitting Ising models for P[S]: results look like SK models -- the method: inversion of TAP equations -- higher-order couplings not necessary -- standard deviations of Jij’s ~same for both kinds of input -- no spin glass phase for these data These hold for the true N (full size of the network). But if we assume a smaller N, we will get a larger time (ms) Comparison of statistics of Jij extracted by algorithm with SK model predictions (stimulus driven network data): time (ms) time (ms) Distributions of Jij’s: tonic and stimulus-driven cases Cross-correlation coefficients: (ni = 1 if neuron i spikes, 0 otherwise) cf Tkacik et al: (10-ms bins) tonic firing: with rapidly- varying stimulus
