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Dynamics of Balanced Networks with Excess Bidirectional Connectivity
Carl van Vreeswijk, Shrisha Rao and David Hansel
Center of Neurophysics, Physiology and Pathology
Cerebral Dynamics, Learning and Memory Laboratory
CNRS-UMR8119 and Paris Descartes University, Paris, France.

There is increasing evidence for fine-structure in cortical connectivity. Bidirectional connectivity and motifs of 3 or more highly interconnected neurons are more prevalent than expected for an Erdös-Rényi random connectivity [1,2]. What is the effect of these excess motifs on cortical dynamics? To address this question we study the dynamics of networks of neurons randomly connected with a rule such that the probability of bidirectional connections is higher than in the pure chance case. The network consists of $NE$ excitatory ($E$) and $NI$ inhibitory ($I$) neurons connected with probability

$Pr ( C ij AB = 1 ) = K / N B , Pr ( C ij AB C ji BA = 1 ) = p AB K / N A N B .$

Here $C AB$ are the connection matrices, $C ij AB = 1$ if there is a connection from neuron $j$ in population $B$ to neuron $i$ in population $A$ and $C ij AB = 0$ otherwise. We focus on the dynamics of such networks operating in the balanced regime [3].

We investigate networks of binary neurons [4] analytically in the limit where we first take $N A → ∞$ and then $K → ∞$, for finite $pAB$. We show that both excess bidirectional connections between the $E$ cells and excess bidirectional connections between the $I$ cells slow down the fluctuations in the neuronal input. As a result, the autocorrelation of the activity decays more slowly than in the corresponding Erdös-Rényi network. In contrast, bidirectional connections between $E$ and $I$ cells decrease the decorrelation time. Remarkably, bidirectional connections between $I$ cells are more efficacious in slowing down the dynamics than those between $E$ cells. These phenomena are due to the small loops that the bidirectionallity induces in the network architecture. Together with the relatively strong synapses in balanced networks these lead to a non-negligible effective delayed self-coupling.

We also investigate the effect of bidirectional connectivity in a balanced network of conductance-based spiking neurons using numerical simulations. Apart from the connectivity, the network is similar to that in [5]. We show that this network behaves qualitatively similarly to the binary network. Furthermore, bidirectional connections between $E$ cells or between $I$ cells increase the Fano factor of the spike count, while the Fano factor decreases for bidirectional connections between $E$ and $I$ cells. We also investigate the dependence of this effect on the synaptic time constants and study how the spike irregularity is modified by 'sensory' stimulation of the network.

[1] S. Song, P.J. Sjostrom, M. Reigl, S. Nelson, and D.B. Chklovskii . PLoS Biol. 3:e68 (2005).
[2] R. Perin, T.K. Berger and H. Markram, PNAS 108:5419-5424 (2011).
[3] C. van Vreeswijk and H. Sompolinsky, Science 274:1724-1726 (1996).
[4] C. van Vreeswijk and H. Sompolinsky, Neural Comput. 10:1321 - 1371 (1998).
[5] D. Hansel and C. van Vreeswijk, J. Neurosci. 32:4049-4064 (2012).