A neural mass model of spectral responses in electrophysiology

Abstract

We present a neural mass model of steady-state membrane potentials measured with local field potentials or electroencephalography in the frequency domain. This model is an extended version of previous dynamic causal models for investigating event-related potentials in the time-domain. In this paper, we augment the previous formulation with parameters that mediate spike-rate adaptation and recurrent intrinsic inhibitory connections. We then use linear systems analysis to show how the model's spectral response changes with its neurophysiological parameters. We demonstrate that much of the interesting behaviour depends on the non-linearity which couples mean membrane potential to mean spiking rate. This non-linearity is analogous, at the population level, to the firing rate-input curves often used to characterize single-cell responses. This function depends on the model's gain and adaptation currents which, neurobiologically, are influenced by the activity of modulatory neurotransmitters. The key contribution of this paper is to show how neuromodulatory effects can be modelled by adding adaptation currents to a simple phenomenological model of EEG. Critically, we show that these effects are expressed in a systematic way in the spectral density of EEG recordings. Inversion of the model, given such non-invasive recordings, should allow one to quantify pharmacologically induced changes in adaptation currents. In short, this work establishes a forward or generative model of electrophysiological recordings for psychopharmacological studies.

Fig. 1

Synaptic impulse response function, convolved with firing rate to produce a postsynaptic membrane potential, parameter values are presented in Table 1.

Fig. 2

Non-linear sigmoid function for PSP to firing rate conversion, parameter values are presented in Table 1.

Fig. 3

Source model, with layered architecture.

Fig. 4

Phenomenological model of adaptation: increasing the value of a in Eq. (5) shifts the firing rate curve to the right and lowers its gain (as approximated linearly by a tangent to the curve at v = 0).

Fig. 5

Example system’s frequency response on the complex s-plane, given two poles and one zero. The magnitude is represented by the height of the curve along the imaginary axis.

Fig. 6

Sensitivity analysis: a pole depicted in the z-place (left) is characterized using eigenvector solutions (right) in terms of its sensitivity to changes in the parameters.

Fig. 7

(a) The effect of variable excitatory time constant τ_e. (b) The effect of variable inhibitory time constant τ_i. (c) The effect of variable maximum excitatory postsynaptic potential H_e. (d) The effect of variable maximum inhibitory postsynaptic potential H_i. (e) The effect of variable pyramidal–pyramidal connection strength γ₁. (f) The effect of variable stellate–pyramidal connection strength γ₂. (g) The effect of variable pyramidal–inhibitory interneurons connection strength γ₃. (h) The effect of variable inhibitory interneurons–pyramidal connection strength γ₄. (i) The effect of variable inhibitory–inhibitory connection strength γ₅.

Fig. 7

(a) The effect of variable excitatory time constant τ_e. (b) The effect of variable inhibitory time constant τ_i. (c) The effect of variable maximum excitatory postsynaptic potential H_e. (d) The effect of variable maximum inhibitory postsynaptic potential H_i. (e) The effect of variable pyramidal–pyramidal connection strength γ₁. (f) The effect of variable stellate–pyramidal connection strength γ₂. (g) The effect of variable pyramidal–inhibitory interneurons connection strength γ₃. (h) The effect of variable inhibitory interneurons–pyramidal connection strength γ₄. (i) The effect of variable inhibitory–inhibitory connection strength γ₅.

Fig. 7

(a) The effect of variable excitatory time constant τ_e. (b) The effect of variable inhibitory time constant τ_i. (c) The effect of variable maximum excitatory postsynaptic potential H_e. (d) The effect of variable maximum inhibitory postsynaptic potential H_i. (e) The effect of variable pyramidal–pyramidal connection strength γ₁. (f) The effect of variable stellate–pyramidal connection strength γ₂. (g) The effect of variable pyramidal–inhibitory interneurons connection strength γ₃. (h) The effect of variable inhibitory interneurons–pyramidal connection strength γ₄. (i) The effect of variable inhibitory–inhibitory connection strength γ₅.

Fig. 8

(a) Changing the gain through (left ρ_i = 1.25, right ρ₁ = 4). (b) The effect of varying gain (inset: fft first-order non-analytic kernel using time-domain equation output).