Oscillations are omnipresent in neural population signals, like multi-unit recordings, EEG/MEG, and the local field potential. They have been linked to the population firing rate of neurons, with individual neurons firing in a close-to-irregular fashion at low rates. Using a combination of mean-field and linear response theory we predict the spectra generated in a layered microcircuit model of V1, composed of leaky integrate-and-fire neurons and based on connectivity compiled from anatomical and electrophysiological studies. The model exhibits low- and high-γ oscillations visible in all populations. Since locally generated frequencies are imposed onto other populations, the origin of the oscillations cannot be deduced from the spectra. We develop an universally applicable systematic approach that identifies the anatomical circuits underlying the generation of oscillations in a given network. Based on a theoretical reduction of the dynamics, we derive a sensitivity measure resulting in a frequency-dependent connectivity map that reveals connections crucial for the peak amplitude and frequency of the observed oscillations and identifies the minimal circuit generating a given frequency. The low-γ peak turns out to be generated in a sub-circuit located in layer 2/3 and 4, while the high-γ peak emerges from the inter-neurons in layer 4. Connections within and onto layer 5 are found to regulate slow rate fluctuations. We further demonstrate how small perturbations of the crucial connections have significant impact on the population spectra, while the impairment of other connections leaves the dynamics on the population level unaltered. The study uncovers connections where mechanisms controlling the spectra of the cortical microcircuit are most effective.
Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering glassy magnetism and frustration, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, representation of probability distributions for sampling-based inference, and learning in the central nervous system based on the dynamic and correlation-dependent modification of synaptic connections. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random, and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate nonlinear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.
Synaptic inhibition is the mechanistic backbone of a suite of cortical functions, not the least of which is maintaining overall network stability as well as modulating neuronal gain. Past cortical models have assumed simplified recurrent networks in which all inhibitory neurons are lumped into a single effective pool. In such models the mechanics of inhibitory stabilization and gain control are tightly linked in opposition to one another -meaning high gain coincides with low stability and vice versa. This tethering of stability and response gain restricts the possible operative regimes of the network. However, it is now well known that cortical inhibition is very diverse, with molecularly distinguished cell classes having distinct positions within the cortical circuit. In this study, we analyze populations of spiking neuron models and associated mean-field theories capturing circuits with pyramidal neurons as well as parvalbumin (PV) and somatostatin (SOM) expressing interneurons. Our study outlines arguments for a division of labor within the full cortical circuit where PV interneurons are ideally positioned to stabilize network activity, whereas SOM interneurons serve to modulate pyramidal cell gain. This segregation of inhibitory function supports stable cortical dynamics over a large range of modulation states. Our study offers a blueprint for how to relate the circuit structure of cortical networks with diverse cell types to the underlying population dynamics and stimulus response.
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