Inferring network connectivity using kinetic Ising models

Modelling time-varying interactions in complex systems: the Score Driven Kinetic Ising Model

A common issue when analyzing real-world complex systems is that the interactions between their elements often change over time. Here we propose a new modeling approach for time-varying interactions generalising the well-known Kinetic Ising Model, a minimalistic pairwise constant interactions model which has found applications in several scientific disciplines. Keeping arbitrary choices of dynamics to a minimum and seeking information theoretical optimality, the Score-Driven methodology allows to extract from data and interpret the presence of temporal patterns describing time-varying interactions. We identify a parameter whose value at a given time can be directly associated with the local predictability of the dynamics and we introduce a method to dynamically learn its value from the data, without specifying parametrically the system's dynamics. We extend our framework to disentangle different sources (e.g. endogenous vs exogenous) of predictability in real time, and show how our methodology applies to a variety of complex systems such as financial markets, temporal (social) networks, and neuronal populations.

Modeling functional resting-state brain networks through neural message passing on the human connectome

In this work, we propose a natural model for information flow in the brain through a neural message-passing dynamics on a structural network of macroscopic regions, such as the human connectome (HC). In our model, each brain region is assumed to have a binary behavior (active or not), the strengths of interactions among them are encoded in the anatomical connectivity matrix defined by the HC, and the dynamics of the system is defined by the Belief Propagation (BP) algorithm, working near the critical point of the network. We show that in the absence of direct external stimuli the BP algorithm converges to a spatial map of activations that is similar to the Default Mode Network (DMN) of the brain, which has been defined from the analysis of functional MRI data. Moreover, we use Susceptibility Propagation (SP) to compute the matrix of long-range correlations between the different regions and show that the modules defined by a clustering of this matrix resemble several Resting State Networks (RSN) determined experimentally. Both results suggest that the functional DMN and RSNs can be seen as simple consequences of the anatomical structure of the brain and a neural message-passing dynamics between macroscopic regions. With the new model, we explore predictions on how functional maps change when the anatomical brain network suffers structural alterations, like in Alzheimer's disease and in lesions of the Corpus Callosum. The implications and novel interpretations suggested by the model, as well as the role of criticality, are discussed.

Inferring structural connectivity using Ising couplings in models of neuronal networks

Functional connectivity metrics have been widely used to infer the underlying structural connectivity in neuronal networks. Maximum entropy based Ising models have been suggested to discount the effect of indirect interactions and give good results in inferring the true anatomical connections. However, no benchmarking is currently available to assess the performance of Ising couplings against other functional connectivity metrics in the microscopic scale of neuronal networks through a wide set of network conditions and network structures. In this paper, we study the performance of the Ising model couplings to infer the synaptic connectivity in in silico networks of neurons and compare its performance against partial and cross-correlations for different correlation levels, firing rates, network sizes, network densities, and topologies. Our results show that the relative performance amongst the three functional connectivity metrics depends primarily on the network correlation levels. Ising couplings detected the most structural links at very weak network correlation levels, and partial correlations outperformed Ising couplings and cross-correlations at strong correlation levels. The result was consistent across varying firing rates, network sizes, and topologies. The findings of this paper serve as a guide in choosing the right functional connectivity tool to reconstruct the structural connectivity.

The Ising decoder: reading out the activity of large neural ensembles

The Ising model has recently received much attention for the statistical description of neural spike train data. In this paper, we propose and demonstrate its use for building decoders capable of predicting, on a millisecond timescale, the stimulus represented by a pattern of neural activity. After fitting to a training dataset, the Ising decoder can be applied "online" for instantaneous decoding of test data. While such models can be fit exactly using Boltzmann learning, this approach rapidly becomes computationally intractable as neural ensemble size increases. We show that several approaches, including the Thouless-Anderson-Palmer (TAP) mean field approach from statistical physics, and the recently developed Minimum Probability Flow Learning (MPFL) algorithm, can be used for rapid inference of model parameters in large-scale neural ensembles. Use of the Ising model for decoding, unlike other problems such as functional connectivity estimation, requires estimation of the partition function. As this involves summation over all possible responses, this step can be limiting. Mean field approaches avoid this problem by providing an analytical expression for the partition function. We demonstrate these decoding techniques by applying them to simulated neural ensemble responses from a mouse visual cortex model, finding an improvement in decoder performance for a model with heterogeneous as opposed to homogeneous neural tuning and response properties. Our results demonstrate the practicality of using the Ising model to read out, or decode, spatial patterns of activity comprised of many hundreds of neurons.

Network inference using asynchronously updated kinetic Ising model

Network structures are reconstructed from dynamical data by respectively naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) approximations. TAP approximation adds simple corrections to the nMF approximation, taking into account the effect of the focused spin on itself via its influence on other neighboring spins. For TAP approximation, we use two methods to reconstruct the network: (a) iterative method; (b) casting the inference formula to a set of cubic equations and solving it directly. We investigate inference of the asymmetric Sherrington-Kirkpatrick (aS-K) model using asynchronous update. The solutions of the set of cubic equations depend on temperature T in the aS-K model, and a critical temperature T(c)≈2.1 is found. The two methods for TAP approximation produce the same results when the iterative method is convergent. Compared to nMF, TAP is somewhat better at low temperatures, but approaches the same performance as temperature increases. Both nMF and TAP approximation reconstruct better for longer data length L, but for the degree of improvement, TAP performs better than nMF.