Threshold-Linear Networks as a Ground-Zero Theory for Spiking Models

Interactions between spiking units grouped in distinct excitatory (E) and inhibitory (I) populations can result in a complex dynamical landscape. Part of this complexity, namely dynamics that depend on undelayed connectivities and external inputs, is retained by a simplified model, the threshold-linear network (TLN), which allows to derive fully analytical conditions for the emergence of multistability [1], paradoxical perturbation effects [2,3], and some oscillations [4,5]. These results, however, have only been applied to specific architectures, such as canonical circuits or all-inhibitory networks, and have not been employed to explain the dynamics of spiking models. Here, we describe a procedure to derive a full-fledged spiking network of leaky integrate-and-fire neurons from a TLN, and we compare the two models across different circuits with a small number of E and I populations.

In a basic E-I architecture, we find that the TLN theory allows to closely approximate the mean firing rate of each population and the border between three different regimes: asynchronous-irregular fixed point, PING oscillation, and suppression of the E population. In the first case, we can further set apart the inhibition-stabilized regime, in which stimulation of the I-units leads to a decrease of their firing rate also in the spiking model. In an I-I circuit, instead, the theory correctly predicts the emergence of "perceptual" bistability [6]. Moving forward to an E-I$_1$-I$_2$ network, we map different regions of the parameter space to three independent hypotheses: PING oscillations with a non-oscillatory E-I$_2$ subsystem [7], bistability of PING oscillations [8], and E-I$_1$ vs I$_2$ bistability, used to model hippocampal sharp waves [9]. The conditions for each regime, accurately predicted by the TLN theory, have an intuitive biophysical interpretation. Then, we study lateral inhibition in an E$_1$-E$_2$-I (unspecific inhibition) and an E$_1$-E$_2$-I$_1$-I$_2$ (clustered inhibition, Figure 1) network. In each case, we derive the explicit mathematical conditions for the cross-interactions being competitive or cooperative, and for these networks to be bistable.

This study demonstrates that the dynamical landscape of spiking networks is less unpredictable than what is commonly believed, and that modelers can more rapidly find and more rigorously interpret specific dynamics of interest within their parameter space by exploiting the analytical results from the TLN theory.