Oscillations emerging from noise-driven steady state in networks with electrical synapses and subthreshold resonance

Abstract

Oscillations play a critical role in cognitive phenomena and have been observed in many brain regions. Experimental evidence indicates that classes of neurons exhibit properties that could promote oscillations, such as subthreshold resonance and electrical gap junctions. Typically, these two properties are studied separately but it is not clear which is the dominant determinant of global network rhythms. Our aim is to provide an analytical understanding of how these two effects destabilize the fluctuation-driven state, in which neurons fire irregularly, and lead to an emergence of global synchronous oscillations. Here we show how the oscillation frequency is shaped by single neuron resonance, electrical and chemical synapses.The presence of both gap junctions and subthreshold resonance are necessary for the emergence of oscillations. Our results are in agreement with several experimental observations such as network responses to oscillatory inputs and offer a much-needed conceptual link connecting a collection of disparate effects observed in networks.

Figure 1. Subthreshold and firing rate resonances of single neurons and the ensuing firing activity.

(a) Subthreshold phase-plane in single neurons; subthreshold oscillations can be observed for complex eigenvalues of subthreshold transformation in equation (1). (b) Q-value of firing rate resonance in single neurons in the τ_V, τ_w-phase plane (dashed line in a); sharpest resonances (high Q) values are observed for parameter values corresponding to subthreshold oscillations in a. (c) Resonant frequencies for firing rate (grey) and subthreshold dynamics (black) at τ_w=4.18 ms. (d) Firing rate of single neurons as a function of input current; solid line (theory) and dots (simulations). Parameters as in Table 1.

Figure 2. Inhibitory and excitatory linear firing rate response.

Normalized linear rate response amplitude R(2πf)/R(0) as a function of input frequency. (a) Rate response exhibits a resonance in a population of inhibitory neurons with the threshold model at low-firing rate. The solid lines are the theory and the dots are the simulations for different values of α. (b) Rate response in the excitatory neurons exhibits a low-pass behaviour. (c) Comparison of the rate responses for the theory and different neuron models, where inhibitory neurons fire at high rate (black line: theory, black dots: adaptive threshold neuron, grey dots: aEIF model). Parameters as in Table 1.

Figure 3. Network response amplitude in experiment and theory.

(a) Spike-triggered effects in inhibitory neurons. Postsynaptic spikelet originating from gap-junction coupling (top) and the GABAergic-mediated IPSC (bottom). (b) Simulated network response R_E(2πf)/R_E(0) (normalized) for excitatory and (c) R_I(2πf)/R_I(0) inhibitory stimulation as a function of input frequency f. Parameters are in Table 1 (black: threshold model, grey: aEIF model). (d) Spike per cycle as a function of stimulation frequency (black: threshold model, grey: aEIF model). (e) Experimentally measured network local field potential (LFP) amplitude in response to light activation of inhibitory cells at 40 Hz (values adapted from Supplementary Fig. 9 in ref. 22) in two different network connectivity conditions (after excitation- and subsequent inhibition blockade) relative to the intact network. (f) Simulated network response of aEIF neurons at 40 Hz in the conditions similar to the experimental data (see Table 1). (g) Simulated network response of aEIF neurons at 40 Hz for different settings of gap-junction strength, recurrent connectivity, mean and variance of the drive.

Figure 4. Global self-sustained oscillations and their frequency.

(a) Phase transition to a global self-sustained oscillatory state as a function of gap-junction strength and current input to inhibitory neurons (top) and to excitatory neurons (bottom). Dashed white line indicates the phase transition between irregular and oscillatory state. Grey scale indicates the log network coherence computed numerically (see Methods, parameters in Table 1). Current input to excitatory neurons is less effective in eliciting global oscillations. (b) Spike rasters of the inhibitory neurons in the oscillatory (top, parameter choice indicated by # in a) and in the asynchronous irregular regime (bottom, parameter choice indicated by * in a). (c) Membrane constant as a function of oscillation frequency: for self-sustained global oscillation (grey area), firing rate resonance (dark grey) and subthreshold resonance (black). In all three conditions, oscillatory frequencies are closely related.