1. 42.13 Spike Pattern Distributions for Model Cortical Networks P-8
John Hertz, Niels Bohr Institute and Nordita; Joanna Tyrcha, Stockholm University
N-dependence is qualitatively consistent
with spike patterns being sampled from
a Sherrington-Kirkpatrick (SK) model:
Outline
Algorithm: Inversion of Thouless-
-Anderson-Palmer equations
Cross-correlations between neurons measured in network
simulations and experiments:
-- for stationary balanced network ±0.0328
-- for highly stimulus-driven network ±
-- 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
TAP equations [1977]
(mean field equations with “reaction field” correction)
an infinite-range (mean field) model: Jij and hi normally distributed with
inverse susceptibility
Given Cij and mi calculated from data, solve for {hi, Jij}
has normal and spin-glass phases; our data are in the normal phase. order parameter equations:
(mean magnetization)
(mean square
magnetization)
Get spike data from simulations
of model network
Excitatory
Population
Inhibitory
External
Input
(Exc.)
We do this for subsets of neurons of sizes N =
2 populations: 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
Results:
Typical Jij: tonic and stimulus-driven cases, N=10
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
Response with tonic input with rapidly-varying input
inhibitory
(100)
16.1 Hz
excitatory
(400)
7.9 Hz
inhibitory
(100)
15.1 Hz
excitatory
(400)
8.6 Hz
Tonic and stimulus J’s very similar
These hold for the true N (full size of the network).
But if we assume a smaller N, we will get a larger E[Jij] and var(Jij)
Comparison of statistics of Jij extracted by algorithm with SK model predictions (stimulus driven network data):
Modeling the distribution:
J’s for a set of neurons ~ J’s within this set obtained
from data from a larger set, except for overall magnitude
Have sets of spike patterns {Si}k (temporal order irrelevant)
Si = ±1 for spike/no spike (we use 10-ms bins)
Construct a distribution P[S] that generates the observed patterns
(i.e., has the same correlations)
Simplest nontrivial model (Schneidman et al, Nature
(2006), Tkačik et al, arXiv:q-bio.NC/ ):
Distributions of Jij’s: tonic and stimulus-driven cases
cf Tkacik et al:
Spin glass phase?
Criterion for normal phase:
J2S < 1 always satisfied
parametrized by Jij, hi; partition function Z normalizes distribution