Effects of Hebbian learning on the dynamics and structure of random networks with inhibitory and excitatory neurons

The aim of the present paper is to study the effects of Hebbian learning in random recurrent neural networks with biological connectivity, i.e. sparse connections and separate populations of excitatory and inhibitory neurons. We furthermore consider that the neuron dynamics may occur at a (shorter) time scale than synaptic plasticity and consider the possibility of learning rules with passive forgetting. We show that the application of such Hebbian learning leads to drastic changes in the network dynamics and structure. In particular, the learning rule contracts the norm of the weight matrix and yields a rapid decay of the dynamics complexity and entropy. In other words, the network is rewired by Hebbian learning into a new synaptic structure that emerges with learning on the basis of the correlations that progressively build up between neurons. We also observe that, within this emerging structure, the strongest synapses organize as a small-world network. The second effect of the decay of the weight matrix spectral radius consists in a rapid contraction of the spectral radius of the Jacobian matrix. This drives the system through the "edge of chaos" where sensitivity to the input pattern is maximal. Taken together, this scenario is remarkably predicted by theoretical arguments derived from dynamical systems and graph theory.     

Finding quasi-optimal network topologies for information transmission in active networks

This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.     

Hierarchical genetic algorithm for near optimal feedforward neural network design

In this paper, we propose a genetic algorithm based design procedure for a multi layer feed forward neural network. A hierarchical genetic algorithm is used to evolve both the neural networks topology and weighting parameters. Compared with traditional genetic algorithm based designs for neural networks, the hierarchical approach addresses several deficiencies, including a feasibility check highlighted in literature. A multi objective cost function is used herein to optimize the performance and topology of the evolved neural network simultaneously. In the prediction of Mackey Glass chaotic time series, the networks designed by the proposed approach prove to be competitive, or even superior, to traditional learning algorithms for the multi layer Perceptron networks and radial basis function networks. Based upon the chosen cost function, a linear weight combination decision making approach has been applied to derive an approximated Pareto optimal solution set. Therefore, designing a set of neural networks can be considered as solving a two objective optimization problem.     

How affinity influences tolerance in an idiotypic network

Idiotypic network models give one possible justification for the appearance of tolerance for a certain category of cells while maintaining immunization for the others. In this paper, we provide new evidence that the manner in which affinity is defined in an idiotypic network model imposes a definite topology on the connectivity of the potential idiotypic network that can emerge. The resulting topology is responsible for very different qualitative behaviour of the network. We show that using a 2D shape-space model with affinity based on complementary regions, a cluster-free topology results that clearly divides the space into distinct zones; if antigens fall into a zone in which there are no available antibodies to bind to, they are tolerated. On the other hand, if they fall into a zone in which there are highly concentrated antibodies available for binding, then they will be eliminated. On the contrary, using a 2D shape space with an affinity function based on cell similarity, a highly clustered topology emerges in which there is no separation of the space into isolated tolerant and non-tolerant zones. Using a bit-string shape space, both similar and complementary affinity measures also result in highly clustered networks. In the networks whose topologies exhibit high clustering, the tolerant and intolerant zones are so intertwined that the networks either reject all antigen or tolerate all antigen. We show that the distribution and topology of the antibody network defined by the complete set of nodes and links-an autonomous feature of the system-therefore selects which antigens are tolerated and which are eliminated.     

On the performance of single-layered neural networks

This paper studies the performance of single-layered neural networks. This study begins with the performance of single-layered neural networks trained using the outer-product rule. The outer-product rule is a suboptimal learning scheme, resulting under certain assumptions from optimal least-squares training of single-layered neural networks with respect to their analog output. Extensive analysis reveals the improvement on the network performance caused by its optimal least-squares training. The effect of the training scheme on the performance of single-layered neural networks with binary output is exhibited by experimentally comparing the performance of single-layered neural networks trained with respect to their analog and binary output.     

A Model for Improving the Learning Curves of Artificial Neural Networks

In this article, the performance of a hybrid artificial neural network (i.e. scale-free and small-world) was analyzed and its learning curve compared to three other topologies: random, scale-free and small-world, as well as to the chemotaxis neural network of the nematode Caenorhabditis Elegans. One hundred equivalent networks (same number of vertices and average degree) for each topology were generated and each was trained for one thousand epochs. After comparing the mean learning curves of each network topology with the C. elegans neural network, we found that the networks that exhibited preferential attachment exhibited the best learning curves.     

Adaptive Neural Network Structure Optimization Algorithm Based on Dynamic Nodes

Large-scale artificial neural networks have many redundant structures, making the network fall into the issue of local optimization and extended training time. Moreover, existing neural network topology optimization algorithms have the disadvantage of many calculations and complex network structure modeling. We propose a Dynamic Node-based neural network Structure optimization algorithm (DNS) to handle these issues. DNS consists of two steps: the generation step and the pruning step. In the generation step, the network generates hidden layers layer by layer until accuracy reaches the threshold. Then, the network uses a pruning algorithm based on Hebb’s rule or Pearson’s correlation for adaptation in the pruning step. In addition, we combine genetic algorithm to optimize DNS (GA-DNS). Experimental results show that compared with traditional neural network topology optimization algorithms, GA-DNS can generate neural networks with higher construction efficiency, lower structure complexity, and higher classification accuracy.     

Stimulus sampling as an exploration mechanism for fast reinforcement learning

Reinforcement learning in neural networks requires a mechanism for exploring new network states in response to a single, nonspecific reward signal. Existing models have introduced synaptic or neuronal noise to drive this exploration. However, those types of noise tend to almost average out—precluding or significantly hindering learning —when coding in neuronal populations or by mean firing rates is considered. Furthermore, careful tuning is required to find the elusive balance between the often conflicting demands of speed and reliability of learning. Here we show that there is in fact no need to rely on intrinsic noise. Instead, ongoing synaptic plasticity triggered by the naturally occurring online sampling of a stimulus out of an entire stimulus set produces enough fluctuations in the synaptic efficacies for successful learning. By combining stimulus sampling with reward attenuation, we demonstrate that a simple Hebbian-like learning rule yields the performance that is very close to that of primates on visuomotor association tasks. In contrast, learning rules based on intrinsic noise (node and weight perturbation) are markedly slower. Furthermore, the performance advantage of our approach persists for more complex tasks and network architectures. We suggest that stimulus sampling and reward attenuation are two key components of a framework by which any single-cell supervised learning rule can be converted into a reinforcement learning rule for networks without requiring any intrinsic noise source. This work was supported by the Swiss National Science Foundation grant K-32K0-118084.     

Generative models of rich clubs in Hebbian neuronal networks and large-scale human brain networks

Rich clubs arise when nodes that are ‘rich’ in connections also form an elite, densely connected ‘club’. In brain networks, rich clubs incur high physical connection costs but also appear to be especially valuable to brain function. However, little is known about the selection pressures that drive their formation. Here, we take two complementary approaches to this question: firstly we show, using generative modelling, that the emergence of rich clubs in large-scale human brain networks can be driven by an economic trade-off between connection costs and a second, competing topological term. Secondly we show, using simulated neural networks, that Hebbian learning rules also drive the emergence of rich clubs at the microscopic level, and that the prominence of these features increases with learning time. These results suggest that Hebbian learning may provide a neuronal mechanism for the selection of complex features such as rich clubs. The neural networks that we investigate are explicitly Hebbian, and we argue that the topological term in our model of large-scale brain connectivity may represent an analogous connection rule. This putative link between learning and rich clubs is also consistent with predictions that integrative aspects of brain network organization are especially important for adaptive behaviour.     

A Three-Threshold Learning Rule Approaches the Maximal Capacity of Recurrent Neural Networks

Understanding the theoretical foundations of how memories are encoded and retrieved in neural populations is a central challenge in neuroscience. A popular theoretical scenario for modeling memory function is the attractor neural network scenario, whose prototype is the Hopfield model. The model simplicity and the locality of the synaptic update rules come at the cost of a poor storage capacity, compared with the capacity achieved with perceptron learning algorithms. Here, by transforming the perceptron learning rule, we present an online learning rule for a recurrent neural network that achieves near-maximal storage capacity without an explicit supervisory error signal, relying only upon locally accessible information. The fully-connected network consists of excitatory binary neurons with plastic recurrent connections and non-plastic inhibitory feedback stabilizing the network dynamics; the memory patterns to be memorized are presented online as strong afferent currents, producing a bimodal distribution for the neuron synaptic inputs. Synapses corresponding to active inputs are modified as a function of the value of the local fields with respect to three thresholds. Above the highest threshold, and below the lowest threshold, no plasticity occurs. In between these two thresholds, potentiation/depression occurs when the local field is above/below an intermediate threshold. We simulated and analyzed a network of binary neurons implementing this rule and measured its storage capacity for different sizes of the basins of attraction. The storage capacity obtained through numerical simulations is shown to be close to the value predicted by analytical calculations. We also measured the dependence of capacity on the strength of external inputs. Finally, we quantified the statistics of the resulting synaptic connectivity matrix, and found that both the fraction of zero weight synapses and the degree of symmetry of the weight matrix increase with the number of stored patterns.     

A linear model for characterization of synchronization frequencies of neural networks

The synchronization frequency of neural networks and its dynamics have important roles in deciphering the working mechanisms of the brain. It has been widely recognized that the properties of functional network synchronization and its dynamics are jointly determined by network topology, network connection strength, i.e., the connection strength of different edges in the network, and external input signals, among other factors. However, mathematical and computational characterization of the relationships between network synchronization frequency and these three important factors are still lacking. This paper presents a novel computational simulation framework to quantitatively characterize the relationships between neural network synchronization frequency and network attributes and input signals. Specifically, we constructed a series of neural networks including simulated small-world networks, real functional working memory network derived from functional magnetic resonance imaging, and real large-scale structural brain networks derived from diffusion tensor imaging, and performed synchronization simulations on these networks via the Izhikevich neuron spiking model. Our experiments demonstrate that both of the network synchronization strength and synchronization frequency change according to the combination of input signal frequency and network self-synchronization frequency. In particular, our extensive experiments show that the network synchronization frequency can be represented via a linear combination of the network self-synchronization frequency and the input signal frequency. This finding could be attributed to an intrinsically-preserved principle in different types of neural systems, offering novel insights into the working mechanism of neural systems.     

Self-organized neural maps of human protein sequences.

We have recently described a method based on artificial neural networks to cluster protein sequences into families. The network was trained with Kohonen''s unsupervised learning algorithm using, as inputs, the matrix patterns derived from the dipeptide composition of the proteins. We present here a large-scale application of that method to classify the 1,758 human protein sequences stored in the SwissProt database (release 19.0), whose lengths are greater than 50 amino acids. In the final 2-dimensional topologically ordered map of 15 x 15 neurons, proteins belonging to known families were associated with the same neuron or with neighboring ones. Also, as an attempt to reduce the time-consuming learning procedure, we compared 2 learning protocols: one of 500 epochs (100 SUN CPU-hours [CPU-h]), and another one of 30 epochs (6.7 CPU-h). A further reduction of learning-computing time, by a factor of about 3.3, with similar protein clustering results, was achieved using a matrix of 11 x 11 components to represent the sequences. Although network training is time consuming, the classification of a new protein in the final ordered map is very fast (14.6 CPU-seconds). We also show a comparison between the artificial neural network approach and conventional methods of biosequence analysis.     

Robust exponential stabilization of a class of delayed neural networks with reaction-diffusion terms

In this paper, the problem of global robust exponential stabilization for a class of neural networks with reaction-diffusion terms and time-varying delays which covers the Hopfield neural networks and cellular neural networks is investigated. A feedback control gain matrix is derived to achieve the global robust exponential stabilization of the neural networks by using the Lyapunov stability theory, and the stabilization condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. This condition can avoid solving an algebraic Riccati equation. Finally, a numerical simulation illustrates the effectiveness of the results.     

Dendritic normalisation improves learning in sparsely connected artificial neural networks

Artificial neural networks, taking inspiration from biological neurons, have become an invaluable tool for machine learning applications. Recent studies have developed techniques to effectively tune the connectivity of sparsely-connected artificial neural networks, which have the potential to be more computationally efficient than their fully-connected counterparts and more closely resemble the architectures of biological systems. We here present a normalisation, based on the biophysical behaviour of neuronal dendrites receiving distributed synaptic inputs, that divides the weight of an artificial neuron’s afferent contacts by their number. We apply this dendritic normalisation to various sparsely-connected feedforward network architectures, as well as simple recurrent and self-organised networks with spatially extended units. The learning performance is significantly increased, providing an improvement over other widely-used normalisations in sparse networks. The results are two-fold, being both a practical advance in machine learning and an insight into how the structure of neuronal dendritic arbours may contribute to computation.     

Pruning artificial neural networks using neural complexity measures

This paper describes a new method for pruning artificial neural networks, using a measure of the neural complexity of the neural network. This measure is used to determine the connections that should be pruned. The measure computes the information-theoretic complexity of a neural network, which is similar to, yet different from previous research on pruning. The method proposed here shows how overly large and complex networks can be reduced in size, whilst retaining learnt behaviour and fitness. The technique proposed here helps to discover a network topology that matches the complexity of the problem it is meant to solve. This novel pruning technique is tested in a robot control domain, simulating a racecar. It is shown, that the proposed pruning method is a significant improvement over the most commonly used pruning method Magnitude Based Pruning. Furthermore, some of the pruned networks prove to be faster learners than the benchmark network that they originate from. This means that this pruning method can also help to unleash hidden potential in a network, because the learning time decreases substantially for a pruned a network, due to the reduction of dimensionality of the network.     

Non-linear data structure extraction using simple hebbian networks

. We present a class a neural networks algorithms based on simple hebbian learning which allow the finding of higher order structure in data. The neural networks use negative feedback of activation to self-organise; such networks have previously been shown to be capable of performing principal component analysis (PCA). In this paper, this is extended to exploratory projection pursuit (EPP), which is a statistical method for investigating structure in high-dimensional data sets. As opposed to previous proposals for networks which learn using hebbian learning, no explicit weight normalisation, decay or weight clipping is required. The results are extended to multiple units and related to both the statistical literature on EPP and the neural network literature on non-linear PCA. Received: 30 May 1994/Accepted in revised form: 18 November 1994     

A high-order graph generating self-organizing structure

A large class of neural network models have their units organized in a lattice with fixed topology or generate their topology during the learning process. These network models can be used as neighborhood preserving map of the input manifold, but such a structure is difficult to manage since these maps are graphs with a number of nodes that is just one or two orders of magnitude less than the number of input points (i.e., the complexity of the map is comparable with the complexity of the manifold) and some hierarchical algorithms were proposed in order to obtain a high-level abstraction of these structures. In this paper a general structure capable to extract high order information from the graph generated by a large class of self-organizing networks is presented. This algorithm will allow to build a two layers hierarchical structure starting from the results obtained by using the suitable neural network for the distribution of the input data. Moreover the proposed algorithm is also capable to build a topology preserving map if it is trained using a graph that is also a topology preserving map.     

Pseudo-relaxation learning algorithm for complex-valued associative memory

HAM (Hopfield Associative Memory) and BAM (Bidirectinal Associative Memory) are representative associative memories by neural networks. The storage capacity by the Hebb rule, which is often used, is extremely low. In order to improve it, some learning methods, for example, pseudo-inverse matrix learning and gradient descent learning, have been introduced. Oh introduced pseudo-relaxation learning algorithm to HAM and BAM. In order to accelerate it, Hattori proposed quick learning. Noest proposed CAM (Complex-valued Associative Memory), which is complex-valued HAM. The storage capacity of CAM by the Hebb rule is also extremely low. Pseudo-inverse matrix learning and gradient descent learning have already been generalized to CAM. In this paper, we apply pseudo-relaxation learning algorithm to CAM in order to improve the capacity.     

A biologically inspired neural network for dynamic programming

An artificial neural network with a two-layer feedback topology and generalized recurrent neurons, for solving nonlinear discrete dynamic optimization problems, is developed. A direct method to assign the weights of neural networks is presented. The method is based on Bellmann's Optimality Principle and on the interchange of information which occurs during the synaptic chemical processing among neurons. The neural network based algorithm is an advantageous approach for dynamic programming due to the inherent parallelism of the neural networks; further it reduces the severity of computational problems that can occur in methods like conventional methods. Some illustrative application examples are presented to show how this approach works out including the shortest path and fuzzy decision making problems.