Neuron model

In the networks (schematic shown in Figure 1A), neurons are modeled using a modified Hodgkin-Huxley model with the addition of a slow, low-threshold K+ current. Due to this current, individual cells display type 2 phase response dynamics (Stiefel et al., 2008), with spike frequency adaptation and subthreshold, theta band membrane potential oscillations. Consequently the neurons act as resonators rather than integrators. The membrane voltage of each neuron is governed by:

Model network and trajectories of measures of synaptic E and I currents for increasing excitatory synaptic strength, w_E. (A) Schematic of model network consisting of excitatory and inhibitory cells randomly coupled with all excitatory (inhibitory) synaptic strengths set to w_E (w_I). All cells receive oscillatory external drive currents at frequencies between 0 and 40 Hz (green curve) and random noisy current inputs (yellow lightning bolts). (B,C) Total synaptic current (E—I difference) is on the y-axis and E/I ratio is on the x-axis. The inhibitory synaptic strength w_I is fixed at 0.3 mS/cm² while the excitatory synaptic strength, w_E, is increased linearly from 0 to 2 mS/cm². (B) No external oscillatory drive is present. The arrows mark the direction of the evolution of the E-I current measures as a function of increasing w_E. Error bars indicate SE over 10 simulation runs with random initial conditions and different random network realizations. (C) Comparison of trajectories in the absence (0Hz, blue) and presence of external oscillatory drive at resonant frequency of 5 Hz (red) and non-resonant frequency of 40 Hz (yellow). (Inset) The maximum E/I ratio on each trajectory curve under external oscillatory stimulation at different frequencies. In this panel and in subsequent figures, results shown are averages over three simulation runs with random initial conditions and different random network realizations.

Each neuron receives a sub-threshold constant current input, which is sampled from a uniform distribution from IiDC = [-0.8, 0.8] μA/cm². In addition, the whole network is driven with a global sinusoidal current with amplitude A = 0.3 μA/cm² (if not stated otherwise). Thus, the external drive is defined as:

where ω is varied to yield oscillations of 0–60 Hz. Values of IiDC and A are chosen so that all neurons display only sub-threshold membrane oscillations even at the peak of each sinusoidal cycle. Each neuron additionally receives Poisson random noisy current input Iinoise consisting of brief (0.05 ms), square, 30 μA/cm² current pulses, delivered at average frequency of 40 Hz.

Ionic currents are gated as follows. For Na + channels:

where h∞(V)={1+exp[V+53.07.0]}−1 , and τh(V)=0.37+2.78{1+exp[V+40.56.0]}−1 .

The kinetics of the K+ delayed rectifier current are governed by:

with n∞(V)={1+exp[−V−30.010.0]}−1 , and τn(V)=0.37+1.85{1+exp[V+27.015.0]}−1 .

The gating of the slow, low-threshold K+ current evolves as:

with z∞(V)={1+exp[−V−39.05.0]}−1 .

The leak conductance is given by g_L = 0.02 mS/cm². Other parameters are set to C = 1 μF/cm², g_Na = 24.0 mS/cm², g_Kdr = 3.0 mS/cm², V_Na = 55.0mV, V_K = -90.0mV, and V_L = -60.0 mV.