The dynamic brain: from spiking neurons to neural masses and cortical fields

Abstract

The cortex is a complex system, characterized by its dynamics and architecture, which underlie many functions such as action, perception, learning, language, and cognition. Its structural architecture has been studied for more than a hundred years; however, its dynamics have been addressed much less thoroughly. In this paper, we review and integrate, in a unifying framework, a variety of computational approaches that have been used to characterize the dynamics of the cortex, as evidenced at different levels of measurement. Computational models at different space-time scales help us understand the fundamental mechanisms that underpin neural processes and relate these processes to neuroscience data. Modeling at the single neuron level is necessary because this is the level at which information is exchanged between the computing elements of the brain; the neurons. Mesoscopic models tell us how neural elements interact to yield emergent behavior at the level of microcolumns and cortical columns. Macroscopic models can inform us about whole brain dynamics and interactions between large-scale neural systems such as cortical regions, the thalamus, and brain stem. Each level of description relates uniquely to neuroscience data, from single-unit recordings, through local field potentials to functional magnetic resonance imaging (fMRI), electroencephalogram (EEG), and magnetoencephalogram (MEG). Models of the cortex can establish which types of large-scale neuronal networks can perform computations and characterize their emergent properties. Mean-field and related formulations of dynamics also play an essential and complementary role as forward models that can be inverted given empirical data. This makes dynamic models critical in integrating theory and experiments. We argue that elaborating principled and informed models is a prerequisite for grounding empirical neuroscience in a cogent theoretical framework, commensurate with the achievements in the physical sciences.

Figure 1. Anatomical connectivity, W = W_hom+W_het, comprising homogeneous and heterogeneous connections.

The intracortical connections are illustrated as densely connected fibers in the upper sheet and define the homogeneous connectivity W_hom. A single fiber connects the two distant regimes (A) and (B) and contributes to the heterogeneous connectivity, W_het, whereas regime (C) has only homogeneous connections.

Figure 2. Typical homogeneous connectivity kernels, W_hom(z), used for local architectures plotted as a function of spatial distance z.

Purely excitatory connectivity is plotted in (A); purely inhibitory in (B); center-on, surround-off in (C); and center-off, surround-on in (D). The connectivity kernel in (C) is the most widely used in computational neuroscience.

Figure 3. Minimal stable regions for the equilibrium state of a neural field as a function of its connectivity and time delay τ = d/c.

The critical surface, at which the equilibrium state undergoes an instability, is plotted as a function of the real and imaginary part of the eigenvalue of its connectivity, W. Regimes below the surface indicate stability, above instability. The vertical axis shows the time delay via transmission along the heterogeneous fiber.

Figure 4. Summary of the stability changes of a neural field with mixed (local/global) connectivity.

(Top) The relative size of stability area for different connectivity kernels. (Bottom) Illustration of change of stability as a function of various factors. Gradient within the arrows indicates the increase of the parameter indicated by each arrow. The direction of the arrow refers to the effect of the related factor on the stability change. The bold line separating stable and unstable regions indicates the course of the critical surface as the time delay changes.

Figure 5. Planar spiking neuron.

(A) Stochastically perturbed fixed point. (B) Limit cycle attractor.

Figure 6. Results of simulating an ensemble of 250 neurons with sensory evoked synaptic currents to all neurons between t = 1,000 ms and t = 3,000 ms.

(A) Raster plot. (B) Mean synaptic currents. (C) Time series of a single neuron. The effect of the input is to effect a bifurcation in each neuron from stochastic to limit cycle dynamics (phase locking), suppressing the impact of the spatially uncorrelated stochastic inputs. As evident in (A), the increased firing synchrony leads in turn to a marked increase in the simulated local field potentials. The mean synaptic currents evidence an emergent phenomenon, and not merely the superposition of a bursting neuron, as can be seen in (C): clearly no burst is evident at this scale.

Figure 7. Contraction of spike-timing differences due to synaptic inputs.

A seed neuron is chosen at random and the interneuron spike difference for all other neurons is plotted each time it spikes. (A) Solid and dashed lines show ±1 and ±1.5 standard deviations of the ensemble spike timing. (B) The normalized fourth moment (excess kurtosis) derived from a moving frame.

Figure 8. Contraction of spike-timing differences due to synaptic inputs.

The left column shows the ensemble state, prior to the stimulus current. The right column shows the intrastimulus activity. Top row: First return map for the cloud interspike delay over five consecutive time steps, before (A) and following (B) synaptic input. The plots are normalized to the average firing rate to control for changes in spike rate. Values for the seed neuron used in Figure 7 are plotted in red. Lower row (C,D): the corresponding spike timing histograms. The ensemble kurtosis increases markedly from sub- to super-Gaussian.

Figure 9. Mesoscopic neural mass model with sensory evoked synaptic currents from t = 1,000 ms to t = 3,000 ms.

Prior to the stimulus, the system is in a stable fixed point regimen. The stochastic inputs act as perturbations around this point. Although individual neurons exhibit nonlinear dynamics, the ensemble mean dynamics are (linearly) stable to the stochastic inputs until the background current is increased. Then the fixed point state is rendered unstable by the stimulus current and large amplitude oscillations occur. These cease following stimulus termination.

Figure 10. Coupled mesoscopic neural masses with sensory evoked synaptic currents into single sensory node from t = 1,000 ms to t = 3,000 ms C_sens>C_sheet.

(A) Individual mean synaptic currents of all nonsensory nodes. (B) Total synaptic currents averaged across the nonsensory sheet. Stimulus-evoked activity from the sensory node reorganizes this activity from spatially incoherent to synchronized.

Figure 11. Coupled mesoscopic neural masses with sensory evoked synaptic currents into single sensory node from t = 1,000 ms to t = 3,000 ms C_sens≈C_sheet.

All other parameters as in Figure 10. (A) Individual mean synaptic currents of all nonsensory nodes. (B) Total synaptic currents averaged across the nonsensory sheet. The injection of the externally evoked sensory currents into the prior activity actually has a slightly desynchronizing effect.

Figure 12. The spatiotemporal evolution of the evoked response of excitatory pulse densities μ(r,t) in an example of sensory-evoked activity (Aquino et. al., unpublished data).

The smooth spatiotemporal dispersion of the evoked cortical response and its time delayed corticothalamic volley are evident.

Figure 13. Decision-making neuronal network.

Minimal neurodynamical model for a probabilistic decision-making network that performs the comparison of two mechanical vibrations applied sequentially (f1 and f2). The model implements a dynamical competition between different neurons. The network contains excitatory pyramidal cells and inhibitory interneurons. The neurons are fully connected (with synaptic strengths as specified in the text). Neurons are clustered into populations. There are two different types of population: excitatory and inhibitory. There are two subtypes of excitatory population, namely: specific and nonselective. Specific populations encode the result of the comparison process in the two-interval vibrotactile discrimination task, i.e., if f1>f2 or f1

Figure 14. Average firing rate of a neuron as a function of f1 and f2, obtained with the spiking simulations of the response of VPC neurons during the comparison period (to be contrasted with the experimental results shown in Figure 2 of [121]).

Diamond points correspond to the average values over 200 trials, and the error bars to the standard deviation. The lines correspond to the mean-field calculations: the black line indicates f1f2 (f2 = f1−8 Hz). The average firing rate of the population f1

Figure 15. Cortical architecture of the model.

The neural field is illustrated by the rectangular box showing the neural activity μ(x,t) composed of inhibitory and excitatory neurons. The input s(x,t) is provided at locations x_i via the Gaussian localization function with width . The explicit model parameters used in the simulations are given in .

Figure 16. Bistable regime of auditory streaming.

The stimulus sequences (top) and its resulting neural field dynamics (bottom).

Figure 17. Percept formation.

For multiple initial conditions, the time series of y(t) are plotted for the three regimes, one stream only (top), bistable (middle), and two streams only (bottom).

Figure 18. van Noorden's bifurcation diagram.

Computational simulations yield the van Noorden's bifurcation diagram as a function of the frequency difference Δf and the IOI. The parameter space is partitioned into three regimes, one region with the percept one stream, another region with the percept two streams and a region in between which permits both.

Figure 19. Crossing scales phenomenon.

The input tone sequences used to form a percept of crossover are shown in (A). The resulting contours of neural field activity are plotted in (B), together with the final time series of y(t) shown in (C). In this particular case, the classification system y(t) does traverse from the positive (two streams) to the negative (one stream) fixed point and back. This trajectory is identified with the percept of crossover. In the case of the bouncing percept, the time series of y(t) will not cross the x-axis as shown in (D).

Figure 20. Simulated nonlinear oscillations arising from 3 Hz modal instability in a corticothalamic neural field model.

(A) Stochastic activity either side of the seizure can be seen, reflecting the response properties of the stable steady state mode. The large amplitude oscillations arise from a transient change in a corticothalamic state parameter from t = 5 s to t = 20 s. (B) Shows more detail of the aperiodic oscillations.

Figure 21. Spatiotemporal activity during 3 Hz spike and wave seizures.

(Left) Shows analysis of an Absence seizure in an adolescent male subject. (Right) Shows the results of a simulated seizure in the corticothalamic field model described in the Recent Developments in Neural Field Models section. Top row shows the evolution of the center frequency of the dominant nonlinear mode. Lower row show the amplitude envelope of these modes. Frequencies and amplitudes were derived using a complex Morlet wavelet decomposition of the real and simulated time series.