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 hyperexcitable neurons within the network can significantly increase the network's computational capacity by bringing it into the chaotic regime.

FIG. 1

Spectra and dynamics of networks with cell-type dependent connectivity (N = 2500). The support of the spectrum of the connectivity matrix J is accurately described by $\sqrt{{\mathrm{\Lambda }}_1}$ (radius of blue circle) for different networks. Top insets - the synaptic gain matrix G summarizes the connectivity structure. Bottom insets - activity of representative neurons from each type. The line ℜ{λ} = 1 (purple) marks the transition from quiescent to chaotic activity. (a) An example chaotic network with two cell-types. The average synaptic gain ḡ (radius of red circle) incorrectly predicts this network to be quiescent. (b) An example silent network. Here ḡ incorrectly predicts this network to be chaotic. (c) An example network with six cell-types. In all examples the radial part of the eigenvalue distribution ρ(|λ|) (orange line) is not uniform [22].

FIG. 2

Autocorrelation modes. Example networks (N = 1200) have 3 equally sized groups with α, g such that M is symmetric. (a) When D* = 1 autocorrelations maintain a constant ratio independent of τ. (b) Rescaling by the components $u_{1c}^R$ collapses the autocorrelation functions (Here Λ₁ = 20, Λ₂ = 0.2, Λ₃ = 0.1). (c) When D* = 2, the autocorrelation functions are linear combinations of two autocorrelation “modes” that decay on different timescales. Projections of these functions $\langle u_c^R|\mathbf{\Delta }(\tau )\rangle$ are shown in (d). Only projections on $|u_1^R\rangle ,|u_2^R\rangle$ are significantly different from 0 (Here Λ₁ = 20, Λ₂ = 16, Λ₃ = 0.1). Insets show the variance of Δ (τ) projected on $|u_c^R\rangle$ averaged over 20 networks in each setting.

FIG. 3

Learning capacity is primarily determined by $\sqrt{{\mathrm{\Lambda }}_1}$, the effective gain of the network. (a) The learning index for four pure frequency target functions (Ω₀ = π/120) plotted as a function of the radius $r=\sqrt{{\mathrm{\Lambda }}_1}({\alpha }_1,\gamma )$. The training epoch lasted approximately 100 periods of the target signal. Each point is an average over 25 networks with N = 500, ε = 0.2 and different values of α₁ and γ. The line is a moving average of these points for each frequency. (b) The same data averaged over the target frequencies shown as a function of γ and α₁. Contour lines of l_Ω (white) and of $\sqrt{{\mathrm{\Lambda }}_1}$ (black) coincide approximately in the region where l_Ω peaks.