Synchronous and asynchronous bursting states: role of intrinsic neural dynamics

Brain signals such as local field potentials often display gamma-band oscillations (30-70 Hz) in a variety of cognitive tasks. These oscillatory activities possibly reflect synchronization of cell assemblies that are engaged in a cognitive function. A type of pyramidal neurons, i.e., chattering neurons, show fast rhythmic bursting (FRB) in the gamma frequency range, and may play an active role in generating the gamma-band oscillations in the cerebral cortex. Our previous phase response analyses have revealed that the synchronization between the coupled bursting neurons significantly depends on the bursting mode that is defined as the number of spikes in each burst. Namely, a network of neurons bursting through a Ca(2+)-dependent mechanism exhibited sharp transitions between synchronous and asynchronous firing states when the neurons exchanged the bursting mode between singlet, doublet and so on. However, whether a broad class of bursting neuron models commonly show such a network behavior remains unclear. Here, we analyze the mechanism underlying this network behavior using a mathematically tractable neuron model. Then we extend our results to a multi-compartment version of the NaP current-based neuron model and prove a similar tight relationship between the bursting mode changes and the network state changes in this model. Thus, the synchronization behavior couples tightly to the bursting mode in a wide class of networks of bursting neurons.     

Synchronous and asynchronous updating in cellular automata.

We analyze the properties of a synchronous and of various asynchronous methods to iterate cellular automata. Asynchronous methods in which the time variable is not explicitly defined, operate by specifying an updating order of the cells. The statistical properties of this order have significant consequences for the dynamics and the patterns generated by the cellular automata. Stronger correlations between consecutive steps in the updating order result in more, artificial structure in the patterns. Among these step-driven methods, using random choice with replacement to pick the next cell for updating, yields results that are least influenced by the updating method. We also analyse a time-driven method in which the state transitions of single cells are governed by a probability per unit time that determines an exponential distribution of the waiting time until the next transition. The statistical properties of this method are completely independent of the size of the grid. Consecutive updating steps therefore show no correlation at all. The stationary states of a cellular automaton do not depend on whether a synchronous or asynchronous updating method is used. Their basins of attraction might, however, be vastly different under synchronous and asynchronous iteration. Cyclic dynamics occur only with synchronous updating.     

Autoassociative memory design using interconnected generalized brain-state-in-a-box neural networks

A class of interconnected neural networks composed of generalized Brain-State-in-a-Box (gBSB) neural subnetworks is considered. Interconnected gBSB neural network architectures are proposed along with their stability conditions. The design of the interconnected neural networks is reduced to the problem of solving linear matrix inequalities (LMIs) to determine the interconnection parameters. A method for solving LMIs is devised generating the solutions that, in general, are further away from zero than the corresponding solutions obtained using MATLAB's LMI toolbox, thus resulting in stronger interconnections between the subnetworks. The proposed architectures are then used to construct neural associative memories. Simulations are performed to illustrate the results obtained.     

Associative memory design using overlapping decomposition and generalized brain-state-in-a-box neural networks

This paper is concerned with large scale associative memory design. A serious problem with neural associative memories is the quadratic growth of the number of interconnections with the problem size. An overlapping decomposition algorithm is proposed to attack this problem. Specifically, a pattern to be processed is decomposed into overlapping sub-patterns. Then, neural sub-networks are constructed that process the sub-patterns. An error correction algorithm operates on the outputs of each sub-network in order to correct the mismatches between sub-patterns that are obtained from the independent recall processes of individual sub-networks. The performance of the proposed large scale associative memory is illustrated using two-dimensional images. It is shown that the proposed method reduces the computing cost of the design of the associative memories compared with non-interconnected associative memories.     

Synchronous and asynchronous systems of threshold elements

The role of synchronism in systems of threshold elements (such as neural networks) is examined. Some important differences between synchronous and asynchronous systems are outlined. In particular, important restrictions on limit cycles are found in asynchronous systems along with multi-frequency oscillations which do not appear in synchronous systems. The possible role of deterministic chaos in these systems is discussed.     

Towards neuro-inspired symbolic models of cognition: linking neural dynamics to behaviors through asynchronous communications

A computational architecture modeling the relation between perception and action is proposed. Basic brain processes representing synaptic plasticity are first abstracted through asynchronous communication protocols and implemented as virtual microcircuits. These are used in turn to build mesoscale circuits embodying parallel cognitive processes. Encoding these circuits into symbolic expressions gives finally rise to neuro-inspired programs that are compiled into pseudo-code to be interpreted by a virtual machine. Quantitative evaluation measures are given by the modification of synapse weights over time. This approach is illustrated by models of simple forms of behaviors exhibiting cognition up to the third level of animal awareness. As a potential benefit, symbolic models of emergent psychological mechanisms could lead to the discovery of the learning processes involved in the development of cognition. The executable specifications of an experimental platform allowing for the reproduction of simulated experiments are given in “Appendix”.     

Dynamic causal models of neural system dynamics: current state and future extensions

Complex processes resulting from interaction of multiple elements can rarely be understood by analytical scientific approaches alone; additional, mathematical models of system dynamics are required. This insight, which disciplines like physics have embraced for a long time already, is gradually gaining importance in the study of cognitive processes by functional neuroimaging. In this field, causal mechanisms in neural systems are described in terms of effective connectivity. Recently, dynamic causal modelling (DCM) was introduced as a generic method to estimate effective connectivity from neuroimaging data in a Bayesian fashion. One of the key advantages of DCM over previous methods is that it distinguishes between neural state equations and modality-specific forward models that translate neural activity into a measured signal. Another strength is its natural relation to Bayesian model selection (BMS) procedures. In this article, we review the conceptual and mathematical basis of DCM and its implementation for functional magnetic resonance imaging data and event-related potentials. After introducing the application of BMS in the context of DCM, we conclude with an outlook to future extensions of DCM. These extensions are guided by the long-term goal of using dynamic system models for pharmacological and clinical applications, particularly with regard to synaptic plasticity.     

Synchronous versus asynchronous oscillations for antigenically varying Plasmodium falciparum with host immune response

We consider a deterministic intra-host model for Plasmodium falciparum (Pf) malaria infection, which accounts for antigenic variation between n clonal variants of PfEMP1 and the corresponding host immune response (IR). Specifically, the model separates the IR into two components, specific and cross-reactive, respectively, in order to demonstrate that the latter can be a mechanism for the sequential appearance of variants observed in actual Pf infections. We show that a strong variant-specific IR relative to the cross-reactive IR favours the asynchronous oscillations (sequential dominance) over the synchronous oscillations in a number of ways. The decay rate of asynchronous oscillations is smaller than that for the synchronous oscillations, allowing for the parasite to survive longer. With the introduction of a delay in the stimulation of the IR, we show that only a small delay is necessary to cause persistent asynchronous oscillations and that a strong variant-specific IR increases the amplitude of the asynchronous oscillations.     

Synchronous versus asynchronous oscillations for antigenically varying Plasmodium falciparum with host immune response

We consider a deterministic intra-host model for Plasmodium falciparum (Pf) malaria infection, which accounts for antigenic variation between n clonal variants of PfEMP1 and the corresponding host immune response (IR). Specifically, the model separates the IR into two components, specific and cross-reactive, respectively, in order to demonstrate that the latter can be a mechanism for the sequential appearance of variants observed in actual Pf infections. We show that a strong variant-specific IR relative to the cross-reactive IR favours the asynchronous oscillations (sequential dominance) over the synchronous oscillations in a number of ways. The decay rate of asynchronous oscillations is smaller than that for the synchronous oscillations, allowing for the parasite to survive longer. With the introduction of a delay in the stimulation of the IR, we show that only a small delay is necessary to cause persistent asynchronous oscillations and that a strong variant-specific IR increases the amplitude of the asynchronous oscillations.     

Positional information in neural map development: lessons from the olfactory system

Positional information is fundamental in development. Although molecular gradients are thought to represent positional information in various systems, the molecular logic used to interpret these gradients remains controversial. In the nervous system, sensory maps are formed in the brain based on gradients of axon guidance molecules. However, it remains unclear how axons find their targets based on relative, not absolute, expression levels of axon guidance receptors. No model solely based on axon-target interactions explains this point. Recent studies in the olfactory system suggested that the neural map formation requires axon-axon interactions, which is known as axon sorting. This review discusses how axon-axon and axon-target interactions interpret molecular gradients and determine the axonal projection sites in neural map formation.     

A polar co-ordinate system for positional information in the vertebrate neural retina

Pattern formation in many developing systems has been traditionally understood in terms of a prior signalling of positional information along mutually orthogonal axes, thus setting up a Cartesian co-ordinate grid. Existing data from the vertebrate neural retina has here been reinterpreted in terms of a non-Cartesian system. Results bearing on regeneration and duplication in eye fragments, “axial determination” and polarity alterations in fused eye fragments are considered and interpreted in terms of a polar co-ordinate system. On this model the position of each region of the retina is defined by a co-ordinate (radial) expressing distance from the centre and another (circumferential) expressing position around the circumference. The radial co-ordinate accords with the radial nature of retinal growth and it is possible that the spatially ordered sequence of cell divisions may itself specify the sequence of radial positions. The circumferential co-ordinate, unlike the radial counterpart, is specified in interaction with the extra-ocular tissue such that it may be oriented in the embryo with reference to a primary embryo Cartesian grid. The ability of the model to account for ultrastructural findings, problematic for the Cartesian model, is discussed.     

Propagation of information in MetaNet graph models.

Information flow in metabolic networks has been studied with a graph model which represents the biochemical transformations occurring in the system under investigation. The signal strength, an algebraic expression which estimates the probability that an intermediate metabolite is bound to a given enzyme, has been used to derive the signal transmittance, the fraction of the informational signal at one intermediate that reaches another intermediate. The transmittance has been used to derive the response ratio, the sensitivity of the rate of change of information at one metabolite consequent to a perturbation at another metabolite. Because the graphical representation corresponds to the biochemical events presumed to occur in the network, these quantities can be used to design experiments to confirm or falsify the hypotheses underlying the model and aid in understanding the regulatory properties of the system. The technique is illustrated by an example model, and its predictions are shown to be sensitive to modest structural changes in the network.     

Control of neural information transmission by synaptic dynamics.

The computational processing of a neural system is strongly influenced by the dynamical characteristics of the information transmission between neurons. In this work, the control of neural information transmission by synaptic dynamics is investigated by means of a master-equation-based stochastic model of pre-synaptic release of neurotransmitter-containing vesicles. The model incorporates facilitation of vesicle fusion with the pre-synaptic membrane due to intracellular calcium ions and depletion of readily releasable vesicles. The message to be transmitted is coded by the pre-synaptic firing sequence, and the received signal corresponds to the post-synaptic membrane potential response. At the sending end, the stochastic character of the vesicle release contributes to the entropy of the probability distribution of the number of vesicles released and represents noise with respect to information transmission. At the receiving end, the generation of post-synaptic membrane potentials is influenced by the temporal behaviour of ionic currents and membrane charging and is determined by means of a low-dimensional model. The rate and temporal types of neural coding are compatible with limiting cases of the synaptic information transmission as a function of initial vesicle release probability and pre-synaptic firing rate. The effects of the nonlinear dependencies of the vesicle release probability on intracellular calcium concentration and number of available vesicles are analysed. The model is compared with phenomenological and reduced models, a principal advantage being the capability of also determining fluctuations of dynamic variables Copyright 2002 Academic Press.     

Attractor analysis of asynchronous Boolean models of signal transduction networks

Prior work on the dynamics of Boolean networks, including analysis of the state space attractors and the basin of attraction of each attractor, has mainly focused on synchronous update of the nodes’ states. Although the simplicity of synchronous updating makes it very attractive, it fails to take into account the variety of time scales associated with different types of biological processes. Several different asynchronous update methods have been proposed to overcome this limitation, but there have not been any systematic comparisons of the dynamic behaviors displayed by the same system under different update methods. Here we fill this gap by combining theoretical analysis such as solution of scalar equations and Markov chain techniques, as well as numerical simulations to carry out a thorough comparative study on the dynamic behavior of a previously proposed Boolean model of a signal transduction network in plants. Prior evidence suggests that this network admits oscillations, but it is not known whether these oscillations are sustained. We perform an attractor analysis of this system using synchronous and three different asynchronous updating schemes both in the case of the unperturbed (wild-type) and perturbed (node-disrupted) systems. This analysis reveals that while the wild-type system possesses an update-independent fixed point, any oscillations eventually disappear unless strict constraints regarding the timing of certain processes and the initial state of the system are satisfied. Interestingly, in the case of disruption of a particular node all models lead to an extended attractor. Overall, our work provides a roadmap on how Boolean network modeling can be used as a predictive tool to uncover the dynamic patterns of a biological system under various internal and environmental perturbations.