On the Validity of Neural Mass Models

Abstract

Modeling the dynamics of neural masses is a common approach in the study of neural populations. Various models have been proven useful to describe a plenitude of empirical observations including self-sustained local oscillations and patterns of distant synchronization. We discuss the extent to which mass models really resemble the mean dynamics of a neural population. In particular, we question the validity of neural mass models if the population under study comprises a mixture of excitatory and inhibitory neurons that are densely (inter-)connected. Starting from a network of noisy leaky integrate-and-fire neurons, we formulated two different population dynamics that both fall into the category of seminal Freeman neural mass models. The derivations contained several mean-field assumptions and time scale separation(s) between membrane and synapse dynamics. Our comparison of these neural mass models with the averaged dynamics of the population reveals bounds in the fraction of excitatory/inhibitory neuron as well as overall network degree for a mass model to provide adequate estimates. For substantial parameter ranges, our models fail to mimic the neural network's dynamics proper, be that in de-synchronized or in (high-frequency) synchronized states. Only around the onset of low-frequency synchronization our models provide proper estimates of the mean potential dynamics. While this shows their potential for, e.g., studying resting state dynamics obtained by encephalography with focus on the transition region, we must accept that predicting the more general dynamic outcome of a neural network via its mass dynamics requires great care.

Keywords: Freeman model; leaky integrate and fire; mean field approximation; neural mass model; random graph.

Figure 1

Typical behavior of the LIF network for different fractions of excitatory and inhibitory units. Top panels (a-c) contain raster plots for 10⁴ units, excitatory in blue and inhibitory in green. The bottom panels (d-f) show the corresponding LIF mean-field potential, V(t); (a,d) λ = 0.6, (b,e) λ = 0.75 and (c,e) λ = 0.8; in all cases the overall network degree was set to p = 0.2.

Figure 2

The synchronization degree in the {p, λ} space computed using the Spike-contrast measure (Ciba et al., 2018). A value close to 1 indicates strong synchronization, here particularly pronounced for large λ, whereas a value close to 0 indicates de-synchronized states. This is the case if p is very small, or if the number of inhibitory units exceeds that of the excitatory ones, i.e., if λ is small.

Figure 3

Median frequency of the LIF network's average potential VLIF; several contour lines were added to highlight the increase of the median frequency when increasing λ. When looking also at Figure 2, one can identify two transitions, one from the de-synchronized to a synchronized state (at low frequencies), followed by a second one from low- to high-frequency synchronization.

Figure 4

χ²-statistic computed between the power spectra (χ²(·, ·)) in the {p, λ} space (10⁵ values). (a) χ² between the LIF network and the CFM, χ²(PLIF, PCFM). (b) χ² between the LIF network and the MFM, χ²(PLIF, PMFM). We add several contour lines in white to improve legibility. The dashed-red line indicates the boundaries of significance region with α = 0.01 (conform the χ² distribution): inside the small region encircle by the dashed-red line, the CFM/MFM spectra were not significantly different from the LIF network spectrum.

Figure 5

Time lags and correlation coefficients. (a,b) depict the optimal time lags τ_max, (a) CFM and (b) MFM. We added contour lines (in white) to improve legibility. In (b) there is a change in the time lags when p is sufficiently large for the LIF network to generate spikes. In (c,d) the corresponding correlation coefficients ρ_k(τ_max) between the LIF model and (c) CFM and (d) MFM are shown. The red-dashed lines in panels (c,d) indicate boundaries of significance; α = 0.01 obtained by applying the Fisher transformation to the correlation values (Fisher, 1915). Inside the area defined by the red-dashed line in the synchronized region and the small area in the asynchronous region where p → 0, the time series of the two neural mass models were not significantly different than the LIF mean field.

Figure 6

Diagram of the network of LIF units (Left) together with the external de-correlated input (Right). Blue denotes excitatory, green inhibitory and black excitatory external neurons. The units on the LIF network are connected to each other with probability p independently on their type. The external units are modeled as independent Poisson trains and are connected with the same probability p^(ext) only to excitatory LIF units and are not connected to inhibitory ones.