1. III-28 [122] Spike Pattern Distributions in Model Cortical Networks Joanna Tyrcha, Stockholm University, Stockholm; John Hertz, Nordita, Stockholm/Copenhagen Outline 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[J ij ] > 0, std[J ij ] falls off ~ no spin glass phase for our data Get spike data from simulations of model network 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 Excitatory Population Inhibitory Population External Input (Exc.) Response to tonic input rapidly-varying stimulus response inhibitory (100) 16.1 Hz excitatory (400) 7.9 Hz time (ms) Cross-correlation coefficients: tonic firing: with rapidly- varying stimulus (10-ms bins) Modeling the distribution of spike patterns Have sets of spike patterns {S i } k for spike/no spike (we use 10-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): parametrized by J ij, h i ; partition function Z normalizes distribution Inversion of Thouless-Anderson-Palmer equations: (Tanaka, PRE 58 2302 (1998)) TAP free energy: function of local “magnetizations” m i = E[S i ] : => “TAP equations” Inversion procedure: Estimate m i and C ij from data: m i =, C ij = - m i m j Invert C matrix and solve for J ij in Solve TAP equations for h i : We do this for subsets of neurons of sizes N = 10-800 Distributions of J ij ’s: tonic and stimulus-driven cases cf Tkacik et al: N-dependence is qualitatively consistent with spike patterns being sampled from a Sherrington-Kirkpatrick (SK) model: an infinite-range (mean field) model: J ij and h i are normally distributed with has normal and spin-glass phases; our data are in the normal phase order parameter equations: mean magnetization mean square magnetization Correlation statistics for the SK model mean correlation matrix element: variance of correlation: where Solve for J 0 and J 1 : Mean and variance of J ij ’s are These hold for the true N (full size of the network). But if we assume a smaller N, we will get a larger Comparison of statistics of J ij extracted by algorithm with SK model predictions (stimulus driven network data): A spin glass state? Phase diagram of the SK model: axes: is the total “field” (external + internal) 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 J ij ’s ~same for both kinds of input -- no spin glass phase for these data ( n i = 1 if neuron i spikes, 0 otherwise) The inferred model is not in the spin glass phase for any N studied. external population rate inhibitory (100) 15.1 Hz excitatory (400) 8.6 Hz SG phase normal phase