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A Novel Locally Balanced Dynamical State in Networks With Structural Heterogeneity And Slow Adaptation
Itamar Daniel Landau1,3,4 and Haim Sompolinsky1,2,3,4
1Edmond and Lily Safra Ctr. for Brain Sci., The Hebrew Univ. of Jerusalem, Jerusalem, Israel 2Ctr. for Brain Sci., Harvard Univ., Cambridge, MA 3Max Planck - Hebrew Univ. of Jerusalem Ctr. for Neurosci. 4The Gatsby Program in Theoretical Neurosci., The Hebrew Univ. of Jerusalem

Traditional analysis of cortical network dynamics has treated simplistic structure -- most commonly, model networks have been constructed with a uniform probability of connection between each pair of cell types. Here we characterize some of the salient features of the connectivity of real local cortical circuits, show that they have significant impact on their dynamical state, and present a novel dynamical state with slow adaptation currents.

First, we present estimates from anatomical data in Layer 4 rat barrel cortex of the local connectivity structure. We show that the local network deviates significantly from a simple random graph, in particular the standard deviation in the number of incoming connections is large -- it is of the same order as the mean. Studying spiking-neuron networks, we use mean-field theory as well as simulations to show that networks with such large structural heterogeneity do not admit a global balanced state solution in which the mean excitation and inhibition dynamically balance to yield fluctuation-driven firing. Rather, the input is locally unbalanced, the network fires extremely sparsely with exceedingly long-tailed rate distributions, and active cells fire with high temporal regularity.

We construct a network model with a slow adaptation current in addition to realistic heterogeneous structure. Using mean-field theory, exact in the asymptotic large network limit, we find a novel dynamical state in which the local adaptation current facilitates the dynamic balance of excitation and inhibition at the global network level, thus recovering the emergent irregular-firing fluctuation-driven state of the original balanced state. This dynamical state has novel non-equilibrium dynamics which are governed by the interaction of the long time scale of the adaptation current with the global E-I balancing.

Finally, we simulate the full Layer 4 population with connectivity estimated from anatomical data to show that this adaptation-facilitated local balance generates realistic cortical firing in both spontaneous state and in response to stimulus.

Our work shows that features of the realistic cortical connectivity as estimated from anatomical data have substantial qualitative impact on their dynamic properties, and that global E-I balance can be stabilized by purely local single-cell properties.