Intrinsically-generated fluctuating activity in excitatory-inhibitory networks

Recurrent networks of non-linear units display a variety of dynamical regimes depending on the structure of their synaptic connectivity. A particularly remarkable phenomenon is the appearance of strongly fluctuating, chaotic activity in networks of deterministic, but randomly connected rate units. How this type of intrinsi- cally generated fluctuations appears in more realistic networks of spiking neurons has been a long standing question. To ease the comparison between rate and spiking networks, recent works investigated the dynami- cal regimes of randomly-connected rate networks with segregated excitatory and inhibitory populations, and firing rates constrained to be positive. These works derived general dynamical mean field (DMF) equations describing the fluctuating dynamics, but solved these equations only in the case of purely inhibitory networks. Using a simplified excitatory-inhibitory architecture in which DMF equations are more easily tractable, here we show that the presence of excitation qualitatively modifies the fluctuating activity compared to purely inhibitory networks. In presence of excitation, intrinsically generated fluctuations induce a strong increase in mean firing rates, a phenomenon that is much weaker in purely inhibitory networks. Excitation moreover induces two different fluctuating regimes: for moderate overall coupling, recurrent inhibition is sufficient to stabilize fluctuations, for strong coupling, firing rates are stabilized solely by the upper bound imposed on activity, even if inhibition is stronger than excitation. These results extend to more general network architectures, and to rate networks receiving noisy inputs mimicking spiking activity. Finally, we show that signatures of the second dynamical regime appear in networks of integrate-and-fire neurons

Granger causality and the inverse Ising problem

We study Ising models for describing data and show that autoregressive methods may be used to learn their connections, also in the case of asymmetric connections and for multi-spin interactions. For each link the linear Granger causality is two times the corresponding transfer entropy (i.e. the information flow on that link) in the weak coupling limit. For sparse connections and a low number of samples, the L1 regularized least squares method is used to detect the interacting pairs of spins. Nonlinear Granger causality is related to multispin interactions.Comment: 6 pages and 8 figures. Revised version in press on Physica

The forward approximation as a mean field approximation for the Anderson and Many Body Localization transitions

In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-) or many-body localized phase. This approximation, which consists in summing over the amplitudes of only the shortest paths in the locator expansion, is known to over-estimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is $0.9\%$ off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder $h_c=4.0\pm 0.3$ is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent $\nu=1$. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium

Realizations of infinite products, Ruelle operators and wavelet filters

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function $M$, we show that a resulting infinite product realization obtained from $M$ takes the form of an (infinite-dimensional) Toeplitz operator with a symbol that is a reflection of the initial realization for $M$. 3) Starting with a subclass of rational matrix functions, including scalar-valued corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of $\mathbf L_2(\mathbb R)$ wavelets. 4) We use both the realizations for $M$ and the corresponding infinite product to produce a matrix representation of the Ruelle-transfer operators used in wavelet theory. By matrix representation we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration.Comment: corrected versio