Volume 12 Supplement 1

## Twentieth Annual Computational Neuroscience Meeting: CNS*2011

# Network inference from non-stationary spike trains

- Joanna Tyrcha
^{1}, - Yasser Roudi
^{2, 3}and - John Hertz
^{3, 4}Email author

**12(Suppl 1)**:P150

**DOI: **10.1186/1471-2202-12-S1-P150

© Tyrcha et al; licensee BioMed Central Ltd. 2011

**Published: **18 July 2011

Current approaches to the problem of inferring network connectivity from spike data [1, 2] make a stationarity assumption, which is often not valid. Here we describe a method for inferring both the connectivity of a network in the presence of nonstationarity state and the time-dependent external drive that causes it.

*S*

_{ i }(

*t*,

*r*) = ±1, according to whether neuron

*i*fires or not in time bin

*t*of repetition

*r*of the measurement. We fit these data to the simplest kind of binary stochastic model: At time step

*t*of repetition

*r*, each formal neuron receives a net input,

*H*

_{ i }(

*t*,

*r*) =

*h*

_{ i }(

*t*) + ∑

_{ j }

*J*

_{ ij }

*S*

_{ j }(

*t*,

*r*), and it takes the value +1 at the next step with a probability given by a logistic sigmoidal function 1/[1+exp(-2

*H*

_{ i }(

*t*,

*r*))] of

*H*

_{ i }(

*t*,

*r*). Maximizing the likelihood of the data leads to learning rules

for the model parameters -- the couplings *J*_{
ij
} and external inputs *h*_{
i
}(*t*). For weak coupling and/or densely connected networks, we have developed faster alternative algorithms [3]. These are based on expanding the learning rules around mean-field and TAP [4] equations for *m*_{
i
}(*t*) = ‹*S*_{
i
}(*t*,*r*)›_{
r
}. (TAP equations are a generalization of the usual mean-field equations for highly connected random networks.)

*J*

_{ ij }s found using the nonstationary algorithm plotted against those found using the stationary one, based on spike trains of 40 salamander retinal neurons stimulated by 120 repetitions of a 26.5-second clip from a film.. The mean

*J*

_{ ij }is reduced, from 0.0471 to -0.0028, and the large positive

*J*

_{ ij }s found assuming stationarity are reduced by a facto of 2-3 when nonstationarity is taken into account.

## Declarations

### Acknowledgements

We thank Michael Berry for providing the salamander retinal data.

## Authors’ Affiliations

## References

- Schneidman E, Berry M, Segev R, Bialek W: Weak pairwise correlations imply strongly correlated networks states in a neural population. Nature. 2006, 440: 1007-1012. 10.1038/nature04701.PubMed CentralView ArticlePubMedGoogle Scholar
- Roudi Y, Tyrcha J, Hertz J: Ising model for neural data: Model quality and approximate methods for extracting functional connectivity. Phys Rev E. 2009, 79: 051915-10.1103/PhysRevE.79.051915.View ArticleGoogle Scholar
- Roudi Y, Hertz J: Mean field theory for nonequilibrium network reconstruction. Phys Rev Lett. 2011, 106: 048702-10.1103/PhysRevLett.106.048702.View ArticlePubMedGoogle Scholar
- Thouless DJ, Anderson PW, Palmer RG: Solution of ‘solvable model of a spin glass’. Phil Mag. 1977, 35: 593-601. 10.1080/14786437708235992.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.