Two-state Neurons Research Articles

A stochastic model of input effectiveness during irregular gamma rhythms.

Gamma-band synchronization has been linked to attention and communication between brain regions, yet the underlying dynamical mechanisms are still unclear. How does the timing and amplitude of inputs to cells that generate an endogenously noisy gamma rhythm affect the network activity and rhythm? How does such "communication through coherence" (CTC) survive in the face of rhythm and input variability? We present a stochastic modelling approach to this question that yields a very fast computation of the effectiveness of inputs to cells involved in gamma rhythms. Our work is partly motivated by recent optogenetic experiments (Cardin et al. Nature, 459(7247), 663-667 2009) that tested the gamma phase-dependence of network responses by first stabilizing the rhythm with periodic light pulses to the interneurons (I). Our computationally efficient model E-I network of stochastic two-state neurons exhibits finite-size fluctuations. Using the Hilbert transform and Kuramoto index, we study how the stochastic phase of its gamma rhythm is entrained by external pulses. We then compute how this rhythmic inhibition controls the effectiveness of external input onto pyramidal (E) cells, and how variability shapes the window of firing opportunity. For transferring the time variations of an external input to the E cells, we find a tradeoff between the phase selectivity and depth of rate modulation. We also show that the CTC is sensitive to the jitter in the arrival times of spikes to the E cells, and to the degree of I-cell entrainment. We further find that CTC can occur even if the underlying deterministic system does not oscillate; quasicycle-type rhythms induced by the finite-size noise retain the basic CTC properties. Finally a resonance analysis confirms the relative importance of the I cell pacing for rhythm generation. Analysis of whole network behaviour, including computations of synchrony, phase and shifts in excitatory-inhibitory balance, can be further sped up by orders of magnitude using two coupled stochastic differential equations, one for each population. Our work thus yields a fast tool to numerically and analytically investigate CTC in a noisy context. It shows that CTC can be quite vulnerable to rhythm and input variability, which both decrease phase preference.

Convergence and Stability of Quantized Hopfield Networks Operating in a Fully Parallel Mode

Hopfield neural networks (HNNs) have proven useful in solving optimization problems that require fast response times. However, the original analog model has an extremely high implementation complexity, making discrete implementations more suitable. Previous work has studied the convergence of discrete-time and quantized-neuron models but has limited the analysis to either two-state neurons or serial operation mode. Nevertheless, two-state neurons have poor performance, and serial operation modes lose fast convergence, which is characteristic of analog HNNs. This letter is the first in the field analyzing the convergence and stability of quantized Hopfield networks (QHNs)-with more than two states-operating in fully parallel mode. Moreover, this letter presents some further analysis on the energy minimization of this type of network. The main conclusion drawn is that QHNs operating in fully parallel mode always converge to a stable state or a cycle of length two and any stable state is a local minimum of the energy.

Rate Reduction for Associative Memory Model in Hodgkin–Huxley-Type Network

We proposed a systematic method for reducing Hodgkin–Huxley-type networks to networks of rate equations on the basis of Shriki et al. 's formulation. Our rate model predicts the results of numerical simulations of the Hodgkin–Huxley-type network model very precisely over a broad range of synaptic conductances. However, in the proposed framework, we ad hoc assumed that the firing threshold and the gain of the f – I curve of the Hodgkin–Huxley-type conductance-based model have a second-order dependence on leak conductance. Here, we discuss optimal model selection with respect to the dependence of the threshold and the gain on the f – I curve, using the Akaike information criterion. We then apply our rate reduction method to an associative memory model of Hodgkin–Huxley neurons. Most associative memory models have been studied using two-state neurons or graded-response neurons. We check the correspondence between an associative memory model of Hodgkin–Huxley neurons and that of graded-response neurons, parti...

A neural network model composed of two-state and three-state neurons

A neural network model composed of two-state (1 and -1) and three-state (1, 0 and -1) neurons is proposed. The two-state neurons are connected with the three-state ones only and vice versa. We derive dynamic equations for the model under the assumption of non-symmetrical dilution of connections. A zero-noise phase diagram is obtained and a region in which two fixed point solutions can coexist is found. Basins of attraction for the solutions are also investigated.

Chaos and synchrony in a model of a hypercolumn in visual cortex

Neurons in cortical slices emit spikes or bursts of spikes regularly in response to a suprathreshold current injection. This behavior is in marked contrast to the behavior of cortical neurons in vivo, whose response to electrical or sensory input displays a strong degree of irregularity. Correlation measurements show a significant degree of synchrony in the temporal fluctuations of neuronal activities in cortex. We explore the hypothesis that these phenomena are the result of the synchronized chaos generated by the deterministic dynamics of local cortical networks. A model of a "hypercolumn" in the visual cortex is studied. It consists of two populations of neurons, one inhibitory and one excitatory. The dynamics of the neurons is based on a Hodgkin-Huxley type model of excitable voltage-clamped cells with several cellular and synaptic conductances. A slow potassium current is included in the dynamics of the excitatory population to reproduce the observed adaptation of the spike trains emitted by these neurons. The pattern of connectivity has a spatial structure which is correlated with the internal organization of hypercolumns in orientation columns. Numerical simulations of the model show that in an appropriate parameter range, the network settles in a synchronous chaotic state, characterized by a strong temporal variability of the neural activity which is correlated across the hypercolumn. Strong inhibitory feedback is essential for the stabilization of this state. These results show that the cooperative dynamics of large neuronal networks are capable of generating variability and synchrony similar to those observed in cortex. Auto-correlation and cross-correlation functions of neuronal spike trains are computed, and their temporal and spatial features are analyzed. In other parameter regimes, the network exhibits two additional states: synchronized oscillations and an asynchronous state. We use our model to study cortical mechanisms for orientation selectivity. It is shown that in a suitable parameter regime, when the input is not oriented, the network has a continuum of states, each representing an inhomogeneous population activity which is peaked at one of the orientation columns. As a result, when a weakly oriented input stimulates the network, it yields a sharp orientation tuning. The properties of the network in this regime, including the appearance of virtual rotations and broad stimulus-dependent cross-correlations, are investigated. The results agree with the predictions of the mean field theory which was previously derived for a simplified model of stochastic, two-state neurons. The relation between the results of the model and experiments in visual cortex are discussed.

Master-equation approach to stochastic neurodynamics.

A master-equation approach to the stochastic neurodynamics proposed by Cowan [in Advances in Neural Information Processing Systems 3, edited by R. P. Lippman, J. E. Moody, and D. S. Touretzky (Morgan Kaufmann, San Mateo, 1991), p. 62] is investigated in this paper. We deal with a model neural network that is composed of two-state neurons obeying elementary stochastic transition rates. We show that such an approach yields concise expressions for multipoint moments and an equation of motion. We apply the formalism to a (1+1)-dimensional system. Exact and approximate expressions for various statistical parameters are obtained and compared with Monte Carlo simulations.

A sequential pseudo-inverse learning rule for networks of formal neurons

For networks of formal two-state neurons the sequential learning problem is considered: a set of patterns has been memorized by adjusting a matrix of the synaptic efficacies. Then an extra pattern is presented to the network and should be stored in addition to previous patterns in such a way that the latter are not used. This problem has been solved in the framework of the pseudo-inverse learning rules. The resulting synaptic matrix has the same value as for other variants of the pseudo-inverse learning rule if memorizing of the previous patterns was performed by some pseudo-inverse rule.

Universality in neural networks: the importance of the 'mean firing rate'.

We present a general analysis of highly connected recurrent neural networks which are able to learn and retrieve a finite number of static patterns. The arguments are based on spike trains and their interval distribution and require no specific model of a neuron. In particular, they apply to formal two-state neurons as well as to more refined models like the integrate-and-fire neuron or the Hodgkin-Huxley equations. We show that the mean firing rate defined as the inverse of the mean interval length is the only relevant parameter (apart from the synaptic weights) that determines the existence of retrieval solutions with a large overlap with one of the learnt patterns. The statistics of the spiking noise (Gaussian, Poisson or other) and hence the shape of the interval distribution does not matter. Thus our unifying approach explains why, and when, all the different associative networks which treat static patterns yield basically the same results, i.e., belong to the same universality class.

Associative memory in a network of ‘spiking’ neurons

The Hopfield network provides a simple model of an associative memory in a neuronal structure. It is, however, based on highly artificial assumptions, especially the use of formal two-state neurons or graded-response neurons. The authors address the question of what happens if formal neurons are replaced by a model of ‘spiking’ neurons. They do so in two steps. First, they show how to include refractoriness and noise into a simple threshold model of neuronal spiking. The spike trains resulting from such a model reproduce the distribution of interspike intervals and gain functions found in real neurons. In a second step they connect the model neurons so as to form a large associative memory system. The spike transmission is described by a synaptic kernel which includes axonal delays, ‘Hebbian’ synaptic efficacies, and a realistic postsynaptic response. The collective behaviour of the system is predicted by a set of dynamical equations which are exact in the limit of a large and fully connected network that...

Associative memory in a network of ‘spiking’ neurons

The Hopfield network provides a simple model of an associative memory in a neuronal structure. It is, however, based on highly artificial assumptions, especially the use of formal two-state neurons or graded-response neurons. The authors address the question of what happens if formal neurons are replaced by a model of ‘spiking’ neurons. They do so in two steps. First, they show how to include refractoriness and noise into a simple threshold model of neuronal spiking. The spike trains resulting from such a model reproduce the distribution of interspike intervals and gain functions found in real neurons. In a second step they connect the model neurons so as to form a large associative memory system. The spike transmission is described by a synaptic kernel which includes axonal delays, ‘Hebbian’ synaptic efficacies, and a realistic postsynaptic response. The collective behaviour of the system is predicted by a set of dynamical equations which are exact in the limit of a large and fully connected network that has to store a finite number of patterns. The authors show that in a stationary retrieval state the statistics of the spiking dynamics is completely wiped out and the system reduces to a network of graded-response neurons. In the case of an oscillatory retrieval state, however, the spiking noise and the internal time constants of the neurons become important and determine the behaviour of the system.

Neural networks: A local learning prescription for arbitrary correlated patterns

A simple local learning rule is presented for neural-network models of N two-state neurons. This rule introduces a free parameter k\ensuremath{\in}(1,4]. We prove that the synaptic matrix converges exponentially to a projection one onto the subspace spanned by the prototypes. This learning mechanism allows embedding without error sets of arbitrary correlated patterns (linearly independent or not), provided that each pattern is presented at most n times, where n is the smallest integer greater than ${\mathrm{log}}_{\mathit{k}}$N.

Weight error sensitivity of fixed point attractors in associative memory networks

The sensitivity of Hopfield neural networks, with two-state neurons is investigated. Simple expressions are derived which give the probability that an equilibrium point of the nominal connection matrix remains a fixed point and an attractor, as a function of the relative error in the weights. Such probability decreases as the number of neurons in the network and the number of stored patterns increase.

Two-state representations of three-state neural networks

The authors investigate to what extent the dynamical behaviour of nets consisting of three-state neurons can be realised in nets using two-state neurons. They find that the generalisation from a two-state neuron to a three-state neuron cannot introduce any new behaviour in the deterministic case, but can do so for noisy nets, even in the zero-noise limit. For those noisy three-state nets which have a Hamiltonian description and a two-state representation they investigate the relationship between the three-state Hamiltonian and the two-state Hamiltonian, and find an interesting level-splitting phenomenon.

Associative memory in an analog iterated-map neural network.

The behavior of an analog neural network with parallel dynamics is studied analytically and numerically for two associative-memory learning algorithms, the Hebb rule and the pseudoinverse rule. Phase diagrams in the parameter space of analog gain \ensuremath{\beta} and storage ratio \ensuremath{\alpha} are presented. For both learning rules, the networks have large ``recall'' phases in which retrieval states exist and convergence to a fixed point is guaranteed by a global stability criterion. We also demonstrate numerically that using a reduced analog gain increases the probability of recall starting from a random initial state. This phenomenon is comparable to thermal annealing used to escape local minima but has the advantage of being deterministic, and therefore easily implemented in electronic hardware. Similarities and differences between analog neural networks and networks with two-state neurons at finite temperature are also discussed.

Associative memory network composed of neurons with hysteretic property

Recalling ability of an associative memory network composed of two-state neurons with hysteretic response property is investigated by a statistical neurodynamical method and numerical experiments. For an auto-correlation-type associative memory network, the capacity becomes greater and basins of attractor of memorized patterns are larger when there is a hysteretic property than otherwise.

Neural networks that use three-state neurons

The dynamics of a neural network which uses three-state neurons (1, 0 and -1) is solved exactly in the limit of non-symmetric and highly dilute synapses. Recursion relations for the 'activity' (the fraction of non-zero neurons) and overlap of the network with a given pattern are derived which have three generic kinds of fixed points: a retrieval fixed point, a chaotic fixed point which corresponds to non-zero activity but no overlap, and a 'zero' fixed point where all the neurons go to the 0 state. As the non-retrieval fixed points both have activities different from the retrieval fixed point, one can easily tell whether a pattern has been recovered. An analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams. Optimising the threshold level can be very important, especially when low-activity patterns are stored. A similar analysis can be applied to 'biased' networks using two-state (1,0) neurons. Finally, one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three-state neurons. This property allows 'larger' (higher activity) memories to be naturally constructed out of smaller ones.

Learning of correlated patterns in spin-glass networks by local learning rules.

Two simple storing prescriptions are presented for neural network models of N two-state neurons. These rules are local and allow the embedding of correlated patterns without errors in a network of spin-glass type. Starting from an arbitrary configuration of synaptic bonds, up to N patterns can be stored by successive modification of the synaptic efficacies. Proofs for the convergence are given. Extensions of these rules are possible.