Dynamics of fully connected attractor neural networks near saturation

Modelling the interplay between the CD4[Formula: see text]/CD8[Formula: see text] T-cell ratio and the expression of MHC-I in tumours

Describing the anti-tumour immune response as a series of cellular kinetic reactions from known immunological mechanisms, we create a mathematical model that shows the CD4[Formula: see text]/CD8[Formula: see text] T-cell ratio, T-cell infiltration and the expression of MHC-I to be interacting factors in tumour elimination. Methods from dynamical systems theory and non-equilibrium statistical mechanics are used to model the T-cell dependent anti-tumour immune response. Our model predicts a critical level of MHC-I expression which determines whether or not the tumour escapes the immune response. This critical level of MHC-I depends on the helper/cytotoxic T-cell ratio. However, our model also suggests that the immune system is robust against small changes in this ratio. We also find that T-cell infiltration and the specificity of the intra-tumour TCR repertoire will affect the critical MHC-I expression. Our work suggests that the functional form of the time evolution of MHC-I expression may explain the qualitative behaviour of tumour growth seen in patients.

Boltzmann Machines as Generalized Hopfield Networks: A Review of Recent Results and Outlooks

The Hopfield model and the Boltzmann machine are among the most popular examples of neural networks. The latter, widely used for classification and feature detection, is able to efficiently learn a generative model from observed data and constitutes the benchmark for statistical learning. The former, designed to mimic the retrieval phase of an artificial associative memory lays in between two paradigmatic statistical mechanics models, namely the Curie-Weiss and the Sherrington-Kirkpatrick, which are recovered as the limiting cases of, respectively, one and many stored memories. Interestingly, the Boltzmann machine and the Hopfield network, if considered to be two cognitive processes (learning and information retrieval), are nothing more than two sides of the same coin. In fact, it is possible to exactly map the one into the other. We will inspect such an equivalence retracing the most representative steps of the research in this field.

Stationary-State Statistics of a Binary Neural Network Model with Quenched Disorder

In this paper, we study the statistical properties of the stationary firing-rate states of a neural network model with quenched disorder. The model has arbitrary size, discrete-time evolution equations and binary firing rates, while the topology and the strength of the synaptic connections are randomly generated from known, generally arbitrary, probability distributions. We derived semi-analytical expressions of the occurrence probability of the stationary states and the mean multistability diagram of the model, in terms of the distribution of the synaptic connections and of the external stimuli to the network. Our calculations rely on the probability distribution of the bifurcation points of the stationary states with respect to the external stimuli, calculated in terms of the permanent of special matrices using extreme value theory. While our semi-analytical expressions are exact for any size of the network and for any distribution of the synaptic connections, we focus our study on networks made of several populations, that we term "statistically homogeneous" to indicate that the probability distribution of their connections depends only on the pre- and post-synaptic population indexes, and not on the individual synaptic pair indexes. In this specific case, we calculated analytically the permanent, obtaining a compact formula that outperforms of several orders of magnitude the Balasubramanian-Bax-Franklin-Glynn algorithm. To conclude, by applying the Fisher-Tippett-Gnedenko theorem, we derived asymptotic expressions of the stationary-state statistics of multi-population networks in the large-network-size limit, in terms of the Gumbel (double exponential) distribution. We also provide a Python implementation of our formulas and some examples of the results generated by the code.

Dreaming neural networks: Forgetting spurious memories and reinforcing pure ones

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is α∼0.14, far from the theoretical bound for symmetric networks, i.e. α=1. Inspired by sleeping and dreaming mechanisms in mammal brains, we propose an extension of this model displaying the standard on-line (awake) learning mechanism (that allows the storage of external information in terms of patterns) and an off-line (sleep) unlearning&consolidating mechanism (that allows spurious-pattern removal and pure-pattern reinforcement): this obtained daily prescription is able to saturate the theoretical bound α=1, remaining also extremely robust against thermal noise. The emergent neural and synaptic features are analyzed both analytically and numerically. In particular, beyond obtaining a phase diagram for neural dynamics, we focus on synaptic plasticity and we give explicit prescriptions on the temporal evolution of the synaptic matrix. We analytically prove that our algorithm makes the Hebbian kernel converge with high probability to the projection matrix built over the pure stored patterns. Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in order to ensure such a convergence. Finally, we run extensive numerical simulations (mainly Monte Carlo sampling) to check the approximations underlying the analytical investigations (e.g., we developed the whole theory at the so called replica-symmetric level, as standard in the Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size effects, finding overall full agreement with the theory.

Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model

Bifurcation theory is a powerful tool for studying how the dynamics of a neural network model depends on its underlying neurophysiological parameters. However, bifurcation theory of neural networks has been developed mostly for mean-field limits of infinite-size spin-glass models, for finite-size dynamical systems whose units have a graded, continuous output, and for models with discrete-output neurons that evolve in continuous time. To allow progress on understanding the dynamics of some widely used classes of neural network models with discrete units and finite size, which could not be studied thoroughly with the previous methodology, here we introduced algorithms that perform a semianalytical bifurcation analysis of a finite-size firing-rate neural network model with binary firing rates and discrete-time evolution. In particular, we focus on the case of small networks composed of tens of neurons, to which existing statistical methods are not applicable. Our approach is based on a numerical brute-force search of the stationary and oscillatory solutions of the model, from which we derive analytical expressions of its bifurcation structure by means of the state-to-state transition probability matrix. Our algorithms determine how the network parameters affect the degree of multistability, the emergence and the period of the neural oscillations, and the formation of spontaneous symmetry breaking in the neural populations. While this technique can be applied to networks with arbitrary (generally asymmetric) connectivity matrices, in particular we introduce a highly efficient algorithm for the bifurcation analysis of sparse networks. We also provide some examples of the obtained bifurcation diagrams and a python implementation of the algorithms.

Associative memory model with long-tail-distributed Hebbian synaptic connections

The postsynaptic potentials of pyramidal neurons have a non-Gaussian amplitude distribution with a heavy tail in both hippocampus and neocortex. Such distributions of synaptic weights were recently shown to generate spontaneous internal noise optimal for spike propagation in recurrent cortical circuits. However, whether this internal noise generation by heavy-tailed weight distributions is possible for and beneficial to other computational functions remains unknown. To clarify this point, we construct an associative memory (AM) network model of spiking neurons that stores multiple memory patterns in a connection matrix with a lognormal weight distribution. In AM networks, non-retrieved memory patterns generate a cross-talk noise that severely disturbs memory recall. We demonstrate that neurons encoding a retrieved memory pattern and those encoding non-retrieved memory patterns have different subthreshold membrane-potential distributions in our model. Consequently, the probability of responding to inputs at strong synapses increases for the encoding neurons, whereas it decreases for the non-encoding neurons. Our results imply that heavy-tailed distributions of connection weights can generate noise useful for AM recall.

"If you can't be with the one you love, love the one you're with": how individual habituation of agent interactions improves global utility

Simple distributed strategies that modify the behavior of selfish individuals in a manner that enhances cooperation or global efficiency have proved difficult to identify. We consider a network of selfish agents who each optimize their individual utilities by coordinating (or anticoordinating) with their neighbors, to maximize the payoffs from randomly weighted pairwise games. In general, agents will opt for the behavior that is the best compromise (for them) of the many conflicting constraints created by their neighbors, but the attractors of the system as a whole will not maximize total utility. We then consider agents that act as creatures of habit by increasing their preference to coordinate (anticoordinate) with whichever neighbors they are coordinated (anticoordinated) with at present. These preferences change slowly while the system is repeatedly perturbed, so that it settles to many different local attractors. We find that under these conditions, with each perturbation there is a progressively higher chance of the system settling to a configuration with high total utility. Eventually, only one attractor remains, and that attractor is very likely to maximize (or almost maximize) global utility. This counterintuitive result can be understood using theory from computational neuroscience; we show that this simple form of habituation is equivalent to Hebbian learning, and the improved optimization of global utility that is observed results from well-known generalization capabilities of associative memory acting at the network scale. This causes the system of selfish agents, each acting individually but habitually, to collectively identify configurations that maximize total utility.

Self-consistent signal-to-noise analysis of Hopfield model with unit replacement

The Hopfield model has a storage capacity: the maximum number of memory patterns that can be stably stored. The memory state of this network model disappears if the number of embedded memory patterns is larger than 0.138N, where N is the system size. Recently, it has been shown in numerical simulations that the Hopfield model with a unit replacement process, in which a small number of old units are replaced with new ones at each learning step for embedding a new pattern, can stably retrieve recently embedded memory patterns even if an infinite number of patterns have been embedded. In this paper, we analyze the Hopfield model with the replacement process by utilizing self-consistent signal-to-noise analysis. We show that 3.21 is the minimum number of replaced units at each learning step that avoids an overload evoking disappearance of the memory state when embedding an infinite number of patterns. Furthermore, we show that the optimal number of replaced units at each learning step that maximizes the number of retrievable patterns is 6.95. These critical numbers of replaced units are independent of the system size N. Finally, we compare this model with the Hopfield model with the forgetting process.

Pattern reconstruction and sequence processing in feed-forward layered neural networks near saturation

The dynamics and the stationary states for the competition between pattern reconstruction and asymmetric sequence processing are studied here in an exactly solvable feed-forward layered neural network model of binary units and patterns near saturation. Earlier work by Coolen and Sherrington on a parallel dynamics far from saturation is extended here to account for finite stochastic noise due to a Hebbian and a sequential learning rule. Phase diagrams are obtained with stationary states and quasiperiodic nonstationary solutions. The relevant dependence of these diagrams and of the quasiperiodic solutions on the stochastic noise and on initial inputs for the overlaps is explicitly discussed.

Hopfield model with self-coupling

We have studied analytically the retrieval performance of a Hopfield model in the presence of self-coupling in the synaptic matrix. We find, contrary to expectations from some earlier studies based on the counting of fixed points, that negative self-coupling causes deterioration in the retrieval performance of the network. On the other hand, it is possible to enhance the retrieval performance by having a positive self-coupling of appropriate magnitude.

Macroscopic dynamics in separable neural networks

Parallel dynamics of neural networks with separable coupling is given starting from Coolen-Sherrington theory. Away from saturation, it is shown that this parallel retrieve dynamics is equivalent to the sequential dynamics for finite temperature. But the finite-size effects were found to be governed by a homogeneous Markov process, not by a time-dependent Ornstein-Uhlenbeck process in sequential dynamics.

Chaos in neural networks with a nonmonotonic transfer function

Time evolution of diluted neural networks with a nonmonotonic transfer function is analytically described by flow equations for macroscopic variables. The macroscopic dynamics shows a rich variety of behaviors: fixed-point, periodicity, and chaos. We examine in detail the structure of the strange attractor and in particular we study the main features of the stable and unstable manifolds, the hyperbolicity of the attractor, and the existence of homoclinic intersections. We also discuss the problem of the robustness of the chaos and we prove that in the present model chaotic behavior is fragile (chaotic regions are densely intercalated with periodicity windows), according to a recently discussed conjecture. Finally we perform an analysis of the microscopic behavior and in particular we examine the occurrence of damage spreading by studying the time evolution of two almost identical initial configurations. We show that for any choice of the parameters the two initial states remain microscopically distinct.

Retrieval dynamics in oscillator neural networks

We present an analytical approach that allows us to treat the long-time behavior of the recalling process in an oscillator neural network. It is well known that in coupled oscillatory neuronal systems, under suitable conditions, the original dynamics can be reduced to a simpler phase dynamics. In this description, the phases of the oscillators can be regarded as the timings of the neuronal spikes. To attempt an analytical treatment of the recalling dynamics of such a system, we study a simplified model in which we discretize time and assume a synchronous updating rule. The theoretical results show that the retrieval dynamics is described by recursion equations for some macroscopic parameters, such as an overlap with the retrieval pattern. We then treat the noise components in the local field, which arise from the learning of the unretrieved patterns, as gaussian variables. However, we take account of the temporal correlation between these noise components at different times. In particular, we find that this correlation is essential for correctly predicting the behavior of the retrieval process in the case of autoassociative memory. From the derived equations, the maximal storage capacity and the basin of attraction are calculated and graphically displayed. We also consider the more general case that the network retrieves an ordered sequence of phase patterns. In both cases, the basin of attraction remains sufficiently wide to recall the memorized pattern from a noisy one, even near saturation. The validity of these theoretical results is supported by numerical simulations. We believe that this model serves as a convenient starting point for the theoretical study of retrieval dynamics in general oscillatory systems.

Notions of Associative Memory and Sparse Coding

This paper summarizes associative memory models and sparse representation of memory in these models. Important properties of the associative memory models are their storage capacity, basin of attraction, and the existence of spurious memories. Sparse coding and nonmonotonic output functions are proposed to improve them. Sparsely coded associative memory model has an extremely large storage capacity which diverges as the mean firing rate of memory patterns approaches 0. The storage capacity strongly depends on the shape of the output function as well as the mean firing rate, even in the case of monotonic output functions. Dynamical properties of the model are analyzed by means of a statistical neurodynamical method. We emphasize the necessity of a feedback mechanism to control the mean firing rate in the recall process. Recently, there have been some experimental results suggesting its existence in the brain. On the other hand, it has been shown that the storage capacity can be markedly improved by replacing the usual monotonic output function with a nonmonotonic one. Another remarkable property of the model with the nonmonotonic neurons is that it seems to have no, or almost no, spurious memory. An associative memory model using nonmonotonic modules with a feedforward inhibition is discussed. The modules consist of two types of threshold units, each of which has a different threshold and can be considered as a biologically plausible representation of the nonmonotonic output function. The above model is compared with the monotonic one. The difference in the storage capacity between the two models becomes small when the sparse patterns are stored. Finally, we discuss the biological plausibility of the discussed associative memory models and sparse coding. Copyright 1996 Elsevier Science Ltd.