On how network architecture determines the dominant patterns of spontaneous neural activity

Abstract

In the absence of sensory stimulation, neocortical circuits display complex patterns of neural activity. These patterns are thought to reflect relevant properties of the network, including anatomical features like its modularity. It is also assumed that the synaptic connections of the network constrain the repertoire of emergent, spontaneous patterns. Although the link between network architecture and network activity has been extensively investigated in the last few years from different perspectives, our understanding of the relationship between the network connectivity and the structure of its spontaneous activity is still incomplete. Using a general mathematical model of neural dynamics we have studied the link between spontaneous activity and the underlying network architecture. In particular, here we show mathematically how the synaptic connections between neurons determine the repertoire of spatial patterns displayed in the spontaneous activity. To test our theoretical result, we have also used the model to simulate spontaneous activity of a neural network, whose architecture is inspired by the patchy organization of horizontal connections between cortical columns in the neocortex of primates and other mammals. The dominant spatial patterns of the spontaneous activity, calculated as its principal components, coincide remarkably well with those patterns predicted from the network connectivity using our theory. The equivalence between the concept of dominant pattern and the concept of attractor of the network dynamics is also demonstrated. This in turn suggests new ways of investigating encoding and storage capabilities of neural networks.

Figure 1. Biologically inspired connectivity.

A: Synaptic strengths of an arbitrary neuron located at the center with its neighbors as a function of distance in two dimensions (Gabor kernel). Positive values indicate excitatory connections and negative values indicate inhibitory connections. B: Projection of the synaptic kernel along an axis crossing the center. Excitatory and inhibitory synapses are spatially periodic, interleaved and their strength decays with distance.

Figure 2. Snapshots of spontaneous activity.

The patterns of spontaneous activity display excited (red) and inhibited (blue) spots with respect to the baseline firing rate (green) that evolve in time (red spots can turn blue and vice versa). The whole movie of the spontaneous activity is provided as Movie S1 in Supporting Information.

Figure 3. Properties of the spontaneous activity.

A: Power spectral density of the arbitrarily chosen neural traces (blue, red, green) and the average across all neural traces (black). Neurons have some oscillatory behavior in the low frequency band (<5 Hz). B: The elements of the theoretically predicted covariance matrix and of the estimated covariance matrix coincide remarkable well (blue dots), as shown by a linear regression (red line) that perfectly overlaps with the identity (y = x, black crosses). C: The eigenvalues of both matrices (blue dots) are accordingly highly correlated (regression in red; identitiy in black dashed lines). D: The principal components of both matrices are also highly correlated. Note the pronounced band along the diagonal of their cross-correlation matrix, which indicates high similarity of the predicted and the estimated dominant modes.

Figure 4. Decomposition of the spontaneous activity in dominant modes.

The spontaneous activity can be mathematically described as a linear superposition of spatial modes (principal components) modulated in time. On the left, we compare some predicted spatial modes with the observed ones noting a good agreement overall. The blue traces on the right represent the temporal modulation of each pattern. The eigenvalue associated with the i-th principal component, or equivalently, the mean quadratic amplitude (variance) of that mode in the spontaneous activity is given by λ_i. The relative variance contained in that mode is expressed as a percentage in parentheses.

Figure 5. Coherent behavior and network attractors.

A: Power spectral density of the dominant modes shown in figure 4 in the same order. The dominant modes are clearly oscillatory with at least one preferred frequency. The superposition of the oscillatory modes endows the spontaneous activity with coherent behavior in space and time. B: The normalized projection, R of the chosen modes onto the spontaneous activity yields the instantaneous contribution of each mode. Thus, the spontaneous activity can also be regarded as fluctuations of the network state around the basins of attraction of different attractors (modes). The black dots represent an instantaneous incursion into the basin of attraction of the corresponding mode. The percentage indicates the relative amount of time spent in the corresponding basin of attraction, i.e. the attractor's dwell time (see Materials and Methods ).