Transition to Chaos in Random Networks with Cell-Type-Specific Connectivity

Abstract

In neural circuits, statistical connectivity rules strongly depend on cell-type identity. We study dynamics of neural networks with cell-type specific connectivity by extending the dynamic mean field method, and find that these networks exhibit a phase transition between silent and chaotic activity. By analyzing the locus of this transition, we derive a new result in random matrix theory: the spectral radius of a random connectivity matrix with block-structured variances. We apply our results to show how a small group of hyper-excitable neurons within the network can significantly increase the network’s computational capacity by bringing it into the chaotic regime.

Low-dimensional dynamics of structured random networks

“…In [13] we used the dynamic mean field approach [17, 24, 25] to study the network behavior in the N → ∞ limit. Averaging Eq.…”

“…Therefore Λ 1 = 1 is the critical point of the D > 1 network. Furthermore, the fact that in the D = 1 case the presence of the destabilized fixed point at x = 0 corresponds to a finite mass of the spectral density of J with real part > 1 [17, 26] allowed us to read the radius of the support of the connectivity matrix with D > 1 and identify it as $r=\sqrt{{\Lambda }_1}$ [13]. …”

“…Recently we showed how a certain type of mesoscopic structure can be introduced into the class of random recurrent network models by drawing synaptic weights from a finite number of cell-type-dependent probability distributions [13]. In contrast to networks with a single cell-type [17], these networks can sustain multiple “modes,” characterized in terms of the individual neuron autocorrelation functions.…”

“…Note that by requiring that g is bounded and differentiable on the unit square outside of S 0 we allow the synaptic gain function to be a combination of discrete modulation (e.g., cell-type dependent connectivity for distinct cell types, as in [13]) and of continuous modulation (e.g., networks with heterogeneous and possibly correlated in- and out-degree distributions, as in [18, 19]).…”

“…A more satisfying treatment to this problem is the recent work of Kadmon and Sompolinsky [23] extending our previous work [13]. They studied a network with block structure, where the distribution of elements in each block has a non-zero mean such that, if appropriately defined, the elements in each column can have the same sign.…”