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On Conductance-Based Neural Field Models

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On Conductance-Based Neural Field Models

Dimitris A Pinotsis et al. Front Comput Neurosci.

Abstract

This technical note introduces a conductance-based neural field model that combines biologically realistic synaptic dynamics-based on transmembrane currents-with neural field equations, describing the propagation of spikes over the cortical surface. This model allows for fairly realistic inter-and intra-laminar intrinsic connections that underlie spatiotemporal neuronal dynamics. We focus on the response functions of expected neuronal states (such as depolarization) that generate observed electrophysiological signals (like LFP recordings and EEG). These response functions characterize the model's transfer functions and implicit spectral responses to (uncorrelated) input. Our main finding is that both the evoked responses (impulse response functions) and induced responses (transfer functions) show qualitative differences depending upon whether one uses a neural mass or field model. Furthermore, there are differences between the equivalent convolution and conductance models. Overall, all models reproduce a characteristic increase in frequency, when inhibition was increased by increasing the rate constants of inhibitory populations. However, convolution and conductance-based models showed qualitatively different changes in power, with convolution models showing decreases with increasing inhibition, while conductance models show the opposite effect. These differences suggest that conductance based field models may be important in empirical studies of cortical gain control or pharmacological manipulations.

Keywords: conductance based models; dynamic causal modeling; electrophysiology; mean field modeling; neural field theory.

Figures

Figure 1
Figure 1
A conductance-based neural field model. This schematic summarizes the equations of motion or state equations that specify a conductance based neural field model of a single source. This model contains three populations, each associated with a specific cortical layer. These equations describe changes in expected neuronal states (e.g., voltage or depolarization) that subtend observed local field potentials or EEG signals. These changes occur as a result of propagating pre-synaptic input through synaptic dynamics. Mean firing rates within each layer are then transformed through a non-linear (sigmoid) voltage-firing rate function to provide (pre-synaptic) inputs to other populations. These inputs are weighted by connection strengths and are gated by the states of synaptic ion channels.
Figure 2
Figure 2
Responses to impulses of different amplitudes for convolution (top) and conductance (bottom) based neural mass models. The responses are normalized with respect to the amplitude of each input. The blue lines illustrate responses to small perturbations. The red lines illustrate responses to intermediate sized inputs, where conductance based models show an augmented response, due to their non-linearity. The green lines show responses for larger inputs, where the saturation effects due to the sigmoid activation function are evident.
Figure 3
Figure 3
Impulse response of conductance and convolution field models to inputs of various amplitudes distinguished by different colours as in Figure 2. The system's flow is generated by Equations (3) and (4a) and the model parameters are given in Tables 1, 2. Non-linear effects are more pronounced—with attenuation of the response amplitude, even for intermediate input amplitudes.
Figure 4
Figure 4
Transfer functions associated with a convolution mass model when changing the excitatory time constant and the connection driving the pyramidal cells over a log-scaling range of (−2, 1) x (−2, −1) (from top to bottom and left to right). The image format summarizes the transfer function in terms of its peak frequency. Transfer functions can be regarded as the spectral response that would be seen if the model was driven by independent (white) fluctuations. They are also the Fourier transform of the impulse response functions of the previous figures.
Figure 5
Figure 5
This figure shows the transfer functions of a cortical source described by a conductance mass model. Here, the intrinsic connectivity and excitatory time constant are changed as in Figure 4. Note the alpha and beta peaks that are typical of these models.
Figure 6
Figure 6
Transfer functions associated with a convolution field model. These are equivalent to the transfer functions shown in Figure 4, where we now model spatial propagation effects with a wave equation. Here, one observes the characteristic increase in frequency when the time constants decrease.
Figure 7
Figure 7
This figure shows the changes in the transfer function of a conductance field model. This is the equivalent to the results for the mass model in Figure 5, where we now include spatial propagation effects.
Figure 8
Figure 8
(Top) Mean depolarization of the pyramidal population of the conductance neural mass model as a function of parameter changes. This corresponds to the fixed point around which the transfer functions in Figure 5 were computed.

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