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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 ${N}_{E}$ excitatory ($E$) and ${N}_{I}$ inhibitory ($I$) neurons connected with probability

Here ${C}^{\text{AB}}$ are the connection matrices, ${C}_{\text{ij}}^{\text{AB}}=\mathrm{1}$ if there is a connection from neuron $j$ in population $B$ to neuron $i$ in population $A$ and ${C}_{\text{ij}}^{\text{AB}}=\mathrm{0}$ 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}\to \infty $ and then $\text{K}\to \infty $, for finite ${p}_{\mathrm{AB}}$. 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).