Synchronous chaos and broad band gamma rhythm in a minimal multi-layer model of primary visual cortex

Abstract

Visually induced neuronal activity in V1 displays a marked gamma-band component which is modulated by stimulus properties. It has been argued that synchronized oscillations contribute to these gamma-band activity. However, analysis of Local Field Potentials (LFPs) across different experiments reveals considerable diversity in the degree of oscillatory behavior of this induced activity. Contrast-dependent power enhancements can indeed occur over a broad band in the gamma frequency range and spectral peaks may not arise at all. Furthermore, even when oscillations are observed, they undergo temporal decorrelation over very few cycles. This is not easily accounted for in previous network modeling of gamma oscillations. We argue here that interactions between cortical layers can be responsible for this fast decorrelation. We study a model of a V1 hypercolumn, embedding a simplified description of the multi-layered structure of the cortex. When the stimulus contrast is low, the induced activity is only weakly synchronous and the network resonates transiently without developing collective oscillations. When the contrast is high, on the other hand, the induced activity undergoes synchronous oscillations with an irregular spatiotemporal structure expressing a synchronous chaotic state. As a consequence the population activity undergoes fast temporal decorrelation, with concomitant rapid damping of the oscillations in LFPs autocorrelograms and peak broadening in LFPs power spectra. We show that the strength of the inter-layer coupling crucially affects this spatiotemporal structure. We predict that layer VI inactivation should induce global changes in the spectral properties of induced LFPs, reflecting their slower temporal decorrelation in the absence of inter-layer feedback. Finally, we argue that the mechanism underlying the emergence of synchronous chaos in our model is in fact very general. It stems from the fact that gamma oscillations induced by local delayed inhibition tend to develop chaos when coupled by sufficiently strong excitation.

Figure 1. Schematic drawing of the model hypercolumns.

A: cartoon of the loop circuit among the 6 layers of striate cortex. Thalamo-recipient layers are indicated by pink shading. B: two-rings network, corresponding to a hypercolumn with interacting layers. LGN inputs are weaker toward the lower layer than toward the upper layer. C: the single ring network for each layer of the model hypercolumn. LGN inputs target both excitatory and inhibitory neurons. D: spatial profile of LGN input. E: spatial modulation of the probability of connections between two cells in the same layer, separated by an angular distance . Red line: excitatory connections. Blue line: inhibitory connections. F: spatial modulation of the probability of connections between two cells in different layers, separated by an angular distance . Red line: upper-to-lower layer excitatory connections and lower-to-upper excitatory connections toward excitatory neurons. Magenta line: lower-to-upper layer excitatory connections toward inhibitory neurons. Blue line: lower-to-upper and upper-to-lower layer inhibitory connections.

Figure 2. Response tuning and contrast response.

A: tuning curves for different contrast levels (re-centered average over N_E = 4000 excitatory neurons in upper layer). Solid lines represent Gaussian fits. B: contrast response functions. Blue curve: average over N_I = 1000 inhibitory neurons in the upper layer. Red curve: average over N_E = 4000 excitatory neurons in the upper layer. Solid lines represent hyperbolic ratio fits.

Figure 3. Low contrast dynamics.

Dynamics of the upper layer for the presentation of a 2%-contrast stimulus. A: raster plot of the excitatory population activity and associated time-histogram of the rate of spiking cells. The histogram bar heights denote the fraction of upper layer excitatory cells that fire in the bin. Bin-size is 2 ms. B: spike trains of 6 excitatory cells highly activated by the presented stimulus. C: membrane potential traces for two neurons stimulated simultaneously at close-to-preferred orientation (2 top neurons of Panel B in red and green). D: pairwise correlations between spike trains (left, cyan histogram) and membrane potentials (right, blue histogram) of highly active neurons.

Figure 4. High contrast dynamics.

Dynamics of the upper layer for the presentation of a 95%-contrast stimulus. A: raster plot of the excitatory population activity and associated time-histogram of the rate of spiking cells. The histogram bar heights denote the fraction of upper layer excitatory cells that fire in the bin. Bin-size is 2 ms. B: spike trains of 6 excitatory cells highly activated by the presented stimulus. C: membrane potential traces for two neurons stimulated simultaneously at close-to-preferred orientation (2 top neurons of Panel B in red and green). D: pairwise correlations between spike trains (magenta histogram) and membrane potentials (red histogram) of highly active neurons.

Figure 5. Pairwise crosscorrelations of spike trains and membrane potentials.

Autocorrelograms and pairwise crosscorrelograms of spiking activity and membrane potentials for three upper layer excitatory neurons. A: spiking activity, low contrast, C = 2%. B: membrane potential, low contrast, C = 2%. C: spiking activity, high contrast, C = 95%. D: membrane potential, high contrast, C = 95%. Auto- and crosscorrelograms are normalized (for zero time-lag, autocorrelograms peak at one and crosscorrelograms at the correlation coefficient). The units for the time-lag axis are ms. Colors are as in Figures 3D and 4D. Rows and columns correspond to different neurons. The angular coordinates of the three neurons are 0°, −10° and 10°.

Figure 6. The Measure of synchrony as a function of the contrast and different network size.

A: The synchrony measure, , increases abruptly with the stimulus contrast N = 10000 (solid line) and N = 40000 (dotted line). B: The synchrony measure as a function of the network size for spontaneous activity (zero contrast, grey line), low contrast (blue line) and high contrast (red line). The dashed line corresponds to a power-law decay with exponent −0.5, denoting a regime of asynchronous activity.

Figure 7. The autocorrelograms of the local field potentials.

A–B: low contrast, C = 2%. C–D: high contrast, C = 95%. Scalings of non-normalized autocorrelograms are shown in B and D. In both cases the damping of secondary peaks is faster for larger network sizes. Zero-lag autocorrelation vanishes for large sizes at low contrast but not at high contrasts. Non-normalized autocorrelations are measured in nA ².

Figure 8. Spectral properties of the LFP and MUA for different contrasts.

A: Power spectra for the LFP induced by a stimulus at preferred orientation. Isolated peaks do not appear even for very high contrast stimuli. B: Average coherence spectra between the MUA and the LFP induced at a same location by a stimulus at preferred orientation. MUA-LFP coherence and LFP power are modulated by contrast changes in the same broad frequency range in the gamma band (30–100 Hz).

Figure 9. Effects of the layer decoupling on the dynamics of the hypercolumn.

Changes for decreasing inter-layer coupling and for a stimulus at high contrast with preferred orientation. A: population average peak firing rate for the excitatory neurons in the upper layer. B: synchrony level . C: autocorrelograms of LFPs for intermediate strengths of the inter-layer coupling ( = 0.8, 0.6 and 0.2). D: corresponding LFP power spectra. E: autocorrelograms of LFP for preferred stimulation at high contrast for the case of fully uncoupled layers ( = 0∶0). F: corresponding LFP power spectrum. Spectra are also plotted for lower levels of contrast and are characterized by a narrow peak at a contrast-dependent frequency.

Figure 10. Short-term response.

Population firing responses to repeated presentations of a high contrast stimulus for fully coupled layers (A–B,  = 1) and for fully uncoupled layers (C–D,  = 0). A and C: peristimulus-time (PST) histograms, based on the firing responses of 500 cells to 1000 presentations of stimuli with optimal (or close to optimal) orientation. B and D: examples of upper layer excitatory population responses for three presentations of the same stimulus.

Figure 11. Chaotic sensitivity to a single spike perturbation.

A black triangle denotes the time of a small perturbation to the network dynamics (for 95% of contrast stimulus and fully-coupled layers,  = 1), in which a single spiking event is omitted. Already after the second oscillation cycle, the unperturbed and perturbed population dynamics have diverged, as visualized by the raster plot (A) and the population rate histogram (B) of the upper layer excitatory population. Blue color denotes unperturbed dynamics and red color perturbed dynamics.