Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junctions in the nervous system, a model system consisting of two nearly identical neurons electrotonically coupled is studied. We assume that each neuron discharges a train of impulses or bursts either spontaneously or under constant stimulus via chemical synapses. It is known that not only an electric current but also chemical substances whose molecular weight is about 1000 can pass through the junction of an electrical synapse (gap junction). So, our model system is regarded as a set of non-linear oscillators coupled by diffusion, and it may be described by a system of ordinary differential equations. Neurons are excited constantly when they are stimulated by an electric current above the threshold level. Therefore, we expect Hopf bifurcation to occur at the critical magnitude of a stimulating electric current in the system of differential equations which describes the dynamics of a single neuron. Studying our model system according to the theory of Hopf bifurcation, we found regions of diffusion constants of the electrical junction which give two kinds of periodic solutions. One is the solution where two neurons oscillate in phase synchrony. The other is where two neurons oscillate 180° out of phase. In the case where one neuron is described by the BVP model, the following was found by computer simulation. When the initial difference between the phase of two neurons is small, the two neurons come to oscillate synchronously. If the initial difference is large, however, the two come to be excited alternately. The physiological implications of these results are discussed.Department of Behaviorology, Faculty of Human Sciences     

Synchronization study in ring-like and grid-like neuronal networks

In this paper, we study the synchronization status of both two gap-junction coupled neurons and neuronal network with two different network connectivity patterns. One of the network connectivity patterns is a ring-like neuronal network, which only considers nearest-neighbor neurons. The other is a grid-like neuronal network, with all nearest neighbor couplings. We show that by varying some key parameters, such as the coupling strength and the external current injection, the neuronal network will exhibit various patterns of firing synchronization. Different types of firing synchronization are diagnosed by means of a mean field potential, a bifurcation diagram, a correlation coefficient and the ISI-distance method. Numerical simulations demonstrate that the synchronization status of multiple neurons is much dependent on the network patters, when the number of neurons is the same. It is also demonstrated that the synchronization status of two coupled neurons is similar with the grid-like neuronal network, but differs radically from that of the ring-like neuronal network. These results may be instructive in understanding synchronization transitions in neuronal systems.     

Bifurcation analysis of a neural network model

This paper describes the analysis of the well known neural network model by Wilson and Cowan. The neural network is modeled by a system of two ordinary differential equations that describe the evolution of average activities of excitatory and inhibitory populations of neurons. We analyze the dependence of the model's behavior on two parameters. The parameter plane is partitioned into regions of equivalent behavior bounded by bifurcation curves, and the representative phase diagram is constructed for each region. This allows us to describe qualitatively the behavior of the model in each region and to predict changes in the model dynamics as parameters are varied. In particular, we show that for some parameter values the system can exhibit long-period oscillations. A new type of dynamical behavior is also found when the system settles down either to a stationary state or to a limit cycle depending on the initial point.     

Influence of noise on the function of a “physiological” neural network

A model neural network with stochastic elements in its millisecond dynamics is investigated. The network consists of neuronal units which are modelled in close analogy to physiological neurons. Dynamical variables of the network are the cellular potentials, axonic currents and synaptic efficacies. The dynamics of the synapses obeys a modified Hebbian rule and, as proposed by v. d. Malsburg (1981, 1985), develop on a time scale of a tenth of a second. In a previous publication (Buhmann and Schulten 1986) we have confirmed that the resulting noiseless autoassociative network is capable of the well-known computational tasks of formal associative networks (Cooper 1973; Kohonen et al. 1984, 1981; Hopfield 1982). In the present paper we demonstrate that random fluctuations of the membrane potential improve the performance of the network. In comparison to a deterministic network a noisy neural network can learn at lower input frequencies and with lower average neural firing rates. The electrical activity of a noisy network is very reminiscent of that observed by physiological recordings. We demonstrate furthermore that associative storage reduces the effective dimension of the phase space in which the electrical activity of the network develops.     

Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities

We investigate how synchrony can be generated or induced in networks of electrically coupled integrate-and-fire neurons subject to noisy and heterogeneous inputs. Using analytical tools, we find that in a network under constant external inputs, synchrony can appear via a Hopf bifurcation from the asynchronous state to an oscillatory state. In a homogeneous net work, in the oscillatory state all neurons fire in synchrony, while in a heterogeneous network synchrony is looser, many neurons skipping cycles of the oscillation. If the transmission of action potentials via the electrical synapses is effectively excitatory, the Hopf bifurcation is supercritical, while effectively inhibitory transmission due to pronounced hyperpolarization leads to a subcritical bifurcation. In the latter case, the network exhibits bistability between an asynchronous state and an oscillatory state where all the neurons fire in synchrony. Finally we show that for time-varying external inputs, electrical coupling enhances the synchronization in an asynchronous network via a resonance at the firing-rate frequency.

The Onset and Extinction of Neural Spiking: A Numerical Bifurcation Approach

We study the onset of neural spiking when the equilibrium rest state loses stability by the change of a critical parameter, the applied current. In the case of the well-known Morris-Lecar model, we start from a complete numerical study of the bifurcation diagram in the most relevant two-parameter range. This diagram includes all equilibrium and limit cycle bifurcations, thus correcting and completing earlier studies.We discuss and classify the behavior of the spiking orbits, when increasing or decreasing the applied current. A complete classification can be extracted from the complete bifurcation diagram. It is based on three components: bifurcation type of the equilibrium at the loss of stability, subcritical behavior in the limit of decreasing the applied current and supercritical behavior in the limit of increasing the applied current.     

Controlled shear affinity filtration (CSAF): a new technology for integration of cell separation and protein isolation from mammalian cell cultures

Up-flow anaerobic sludge blanket (UASB) reactors are being used with increasing regularity all over the world, especially in India, for a variety of wastewater treatment operations. Consequently, there is a need to develop methodologies enabling one to determine UASB reactor performance, not only for designing more efficient UASB reactors but also for predicting the performance of existing reactors under various conditions of influent wastewater flows and characteristics. This work explores the feasibility of application of an artificial neural network-based model for simulating the performance of an existing UASB reactor. Accordingly, a neural network model was designed and trained to predict the steady-state performance of a UASB reactor treating high-strength (unrefined sugar based) wastewater. The model inputs were organic loading rate, hydraulic retention time, and influent bicarbonate alkalinity. The output variables were one or more of the following, effluent substrate concentration (Se), reactor bicarbonate alkalinity, reactor pH, reactor volatile fatty acid concentration, average gas production rate, and percent methane content of the gas. Training of the neural network model was achieved using a large amount of experimentally obtained reactor performance data from the reactor mentioned above as the training set. Training was followed by validation using independent sets of performance data obtained from the same UASB reactor. Subsequently, simulations were performed using the validated neural network model to determine the impact of changes in parameters like influent chemical oxygen demand (COD) concentration and hydraulic retention time on the reactor performance. Simulation results thus obtained were carefully analyzed based on qualitative understanding of UASB process and were found to provide important insights into key variables that were responsible for influencing the working of the UASB reactor under varying input conditions.     

Study of synaptic plasticity via random graphs

The dynamical random graphs associated with a certain class of biological neural networks are introduced and studied. We describe the phase diagram revealing the parameters of a single neuron and of the synaptic strengths which allow formation of the stable strongly connected large groups of neurons. It is shown that the cycles are the most stable structures when the Hebb rule is implemented into the dynamics of the network of excitatory neurons. We discuss the role of cycles for the synchronization of the neuronal activity.     

Cut-loading: A Useful Tool for Examining the Extent of Gap Junction Tracer Coupling Between Retinal Neurons

In addition to chemical synaptic transmission, neurons that are connected by gap junctions can also communicate rapidly via electrical synaptic transmission. Increasing evidence indicates that gap junctions not only permit electrical current flow and synchronous activity between interconnected or coupled cells, but that the strength or effectiveness of electrical communication between coupled cells can be modulated to a great extent^1,2. In addition, the large internal diameter (~1.2 nm) of many gap junction channels permits not only electric current flow, but also the diffusion of intracellular signaling molecules and small metabolites between interconnected cells, so that gap junctions may also mediate metabolic and chemical communication. The strength of gap junctional communication between neurons and its modulation by neurotransmitters and other factors can be studied by simultaneously electrically recording from coupled cells and by determining the extent of diffusion of tracer molecules, which are gap junction permeable, but not membrane permeable, following iontophoretic injection into single cells. However, these procedures can be extremely difficult to perform on neurons with small somata in intact neural tissue.Numerous studies on electrical synapses and the modulation of electrical communication have been conducted in the vertebrate retina, since each of the five retinal neuron types is electrically connected by gap junctions^3,4. Increasing evidence has shown that the circadian (24-hour) clock in the retina and changes in light stimulation regulate gap junction coupling^3-8. For example, recent work has demonstrated that the retinal circadian clock decreases gap junction coupling between rod and cone photoreceptor cells during the day by increasing dopamine D2 receptor activation, and dramatically increases rod-cone coupling at night by reducing D2 receptor activation^7,8. However, not only are these studies extremely difficult to perform on neurons with small somata in intact neural retinal tissue, but it can be difficult to adequately control the illumination conditions during the electrophysiological study of single retinal neurons to avoid light-induced changes in gap junction conductance.Here, we present a straightforward method of determining the extent of gap junction tracer coupling between retinal neurons under different illumination conditions and at different times of the day and night. This cut-loading technique is a modification of scrape loading^9-12, which is based on dye loading and diffusion through open gap junction channels. Scrape loading works well in cultured cells, but not in thick slices such as intact retinas. The cut-loading technique has been used to study photoreceptor coupling in intact fish and mammalian retinas^{7, 8,13}, and can be used to study coupling between other retinal neurons, as described here.     

Modelling the Effects of Electrical Coupling between Unmyelinated Axons of Brainstem Neurons Controlling Rhythmic Activity

Gap junctions between fine unmyelinated axons can electrically couple groups of brain neurons to synchronise firing and contribute to rhythmic activity. To explore the distribution and significance of electrical coupling, we modelled a well analysed, small population of brainstem neurons which drive swimming in young frog tadpoles. A passive network of 30 multicompartmental neurons with unmyelinated axons was used to infer that: axon-axon gap junctions close to the soma gave the best match to experimentally measured coupling coefficients; axon diameter had a strong influence on coupling; most neurons were coupled indirectly via the axons of other neurons. When active channels were added, gap junctions could make action potential propagation along the thin axons unreliable. Increased sodium and decreased potassium channel densities in the initial axon segment improved action potential propagation. Modelling suggested that the single spike firing to step current injection observed in whole-cell recordings is not a cellular property but a dynamic consequence of shunting resulting from electrical coupling. Without electrical coupling, firing of the population during depolarising current was unsynchronised; with coupling, the population showed synchronous recruitment and rhythmic firing. When activated instead by increasing levels of modelled sensory pathway input, the population without electrical coupling was recruited incrementally to unpatterned activity. However, when coupled, the population was recruited all-or-none at threshold into a rhythmic swimming pattern: the tadpole “decided” to swim. Modelling emphasises uncertainties about fine unmyelinated axon physiology but, when informed by biological data, makes general predictions about gap junctions: locations close to the soma; relatively small numbers; many indirect connections between neurons; cause of action potential propagation failure in fine axons; misleading alteration of intrinsic firing properties. Modelling also indicates that electrical coupling within a population can synchronize recruitment of neurons and their pacemaker firing during rhythmic activity.     

Synchronization of an embryonic network of identified spinal interneurons solely by electrical coupling

There is a need to understand the mechanisms of neural synchronization during development because correlated rhythmic activity is thought to be critical for the establishment of proper connectivity. The relative importance of chemical and electrical synapses for synchronization of electrical activity during development is unclear. We examined the activity patterns of identified spinal neurons at the onset of motor activity in zebrafish embryos. Rhythmic activity appeared early and persisted upon blocking chemical neurotransmission but was abolished by inhibitors of gap junctions. Paired recordings revealed that active spinal neurons were electrically coupled and formed a simple network of motoneurons and a subset of interneurons. Thus, the earliest spinal central pattern generator consists of synchronously active, electrically coupled neurons.     

Modulation of a feeding neural circuit by microinjection of K+ channel expression genes into a single identified neuron in Aplysia kurodai

In Aplysia buccal ganglion expression genes for voltage-dependent K(+) channels (AKv1.1a) were injected into one of four electrically coupled multi-action (MA) neurons that directly inhibit jaw-closing (JC) motor neurons and may cooperatively generate their firing pattern during the feeding response. Following the DNA injection, the firing threshold increased and the spike frequency at the same current decreased in the current-induced excitation of the MA neuron; indicating a decrease in excitability of the MA neuron. This procedure also reduced the firing activity of MA neurons during the feeding-like rhythmic responses induced by the electrical nerve stimulation. Moreover, the firing pattern in JC motor neurons was remarkably changed, suggesting the effective contribution of a single MA neuron or electrically coupled MA neurons to the generation of the firing pattern in the JC motor neurons. This method appears useful for exploring the functional roles of specific neurons in complex neural circuits.     

The Role of Axonal Delay in the Synchronization of Networks of Coupled Cortical Oscillators

Coupled oscillator models use a single phase variable to approximate the voltage oscillation of each neuron during repetitivefiring where the behavior of the model depends on the connectivityand the interaction function chosen to describe the coupling. Weintroduce a network model consisting of a continuum of theseoscillators that includes the effects of spatially decaying coupling and axonal delay. We derive equations for determining the stability of solutions and analyze the network behavior for two different interaction functions. The first is a sine function, and the second is derived from a compartmental model of a pyramidal cell.In both cases, the system of coupled neural oscillators can undergo a bifurcation from synchronous oscillations to waves.The change in qualitative behavior is due to the axonal delay,which causes distant connections to encourage a phase shift between cells. We suggest that this mechanism could contribute to the behavior observed in several neurobiological systems.     

Biophysical basis for three distinct dynamical mechanisms of action potential initiation

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.     

Travelling waves and EEG patterns during epileptic seizure: analysis with an integrate-and-fire neural network

Epilepsy is characterized by paradoxical patterns of neural activity. They may cause different types of electroencephalogram (EEG), which dynamically change in shape and frequency content during the temporal evolution of seizure. It is generally assumed that these epileptic patterns may originate in a network of strongly interconnected neurons, when excitation dominates over inhibition. The aim of this work is to use a neural network composed of 50 x 50 integrate-and-fire neurons to analyse which parameter alterations, at the level of synapse topology, may induce network instability and epileptic-like discharges, and to study the corresponding spatio-temporal characteristics of electrical activity in the network. We assume that a small group of central neurons is stimulated by a depolarizing current (epileptic focus) and that neurons are connected via a Mexican-hat topology of synapses. A signal representative of cortical EEG (ECoG) is simulated by summing the membrane potential changes of all neurons. A sensitivity analysis on the parameters describing the synapse topology shows that an increase in the strength and in spatial extension of excitatory vs. inhibitory synapses may cause the occurrence of travelling waves, which propagate along the network. These propagating waves may cause EEG patterns with different shape and frequency, depending on the particular parameter set used during the simulations. The resulting model EEG signals include irregular rhythms with large amplitude and a wide frequency content, low-amplitude high-frequency rapid discharges, isolated or repeated bursts, and low-frequency quasi-sinusoidal patterns. A slow progressive temporal variation in a single parameter may cause the transition from one pattern to another, thus generating a highly non-stationary signal which resembles that observed during ECoG measurements. These results may help to elucidate the mechanisms at the basis of some epileptic discharges, and to relate rapid changes in EEG patterns with the underlying alterations at the network level.     

Effects of heterogeneity in synaptic conductance between weakly coupled identical neurons

A significant degree of heterogeneity in synaptic conductance is present in neuron to neuron connections. We study the dynamics of weakly coupled pairs of neurons with heterogeneities in synaptic conductance using Wang–Buzsaki and Hodgkin–Huxley model neurons which have Types I and II excitability, respectively. This type of heterogeneity breaks a symmetry in the bifurcation diagrams of equilibrium phase difference versus the synaptic rate constant when compared to the identical case. For weakly coupled neurons coupled with identical values of synaptic conductance a phase locked solution exists for all values of the synaptic rate constant, α. In particular, in-phase and anti-phase solutions are guaranteed to exist for all α. Heterogeneity in synaptic conductance results in regions where no phase locked solution exists and the general loss of the ubiquitous in-phase and anti-phase solutions of the identically coupled case. We explain these results through examination of interaction functions using the weak coupling approximation and an in-depth analysis of the underlying multiple cusp bifurcation structure of the systems of coupled neurons.     

Does the computer act like the braine, or like any biological cell?

A new conception of computer functioning is proposed, which suggests that a computer is a reactor in which a unique enzymic reaction at a single operation mode occurs. The analogy between informatics and enzymology is considered. In the context of the parallelism proposed, the work of the brain is analogous to the work of a global computer network. Not the brain as a network of neuron processors but a network of computers as the brain!     

Network bursts in cortical cultures are best simulated using pacemaker neurons and adaptive synapses

One of the most specific and exhibited features in the electrical activity of dissociated cultured neural networks (NNs) is the phenomenon of synchronized bursts, whose profiles vary widely in shape, width and firing rate. On the way to understanding the organization and behavior of biological NNs, we reproduced those features with random connectivity network models with 5,000 neurons. While the common approach to induce bursting behavior in neuronal network models is noise injection, there is experimental evidence suggesting the existence of pacemaker-like neurons. In our simulations noise did evoke bursts, but with an unrealistically gentle rising slope. We show that a small subset of ‘pacemaker’ neurons can trigger bursts with a more realistic profile. We found that adding pacemaker-like neurons as well as adaptive synapses yield burst features (shape, width, and height of the main phase) in the same ranges as obtained experimentally. Finally, we demonstrate how changes in network connectivity, transmission delays, and excitatory fraction influence network burst features quantitatively.     

High-Frequency Stimulation of Excitable Cells and Networks

High-frequency (HF) stimulation has been shown to block conduction in excitable cells including neurons and cardiac myocytes. However, the precise mechanisms underlying conduction block are unclear. Using a multi-scale method, the influence of HF stimulation is investigated in the simplified FitzhHugh-Nagumo and biophysically-detailed Hodgkin-Huxley models. In both models, HF stimulation alters the amplitude and frequency of repetitive firing in response to a constant applied current and increases the threshold to evoke a single action potential in response to a brief applied current pulse. Further, the excitable cells cannot evoke a single action potential or fire repetitively above critical values for the HF stimulation amplitude. Analytical expressions for the critical values and thresholds are determined in the FitzHugh-Nagumo model. In the Hodgkin-Huxley model, it is shown that HF stimulation alters the dynamics of ionic current gating, shifting the steady-state activation, inactivation, and time constant curves, suggesting several possible mechanisms for conduction block. Finally, we demonstrate that HF stimulation of a network of neurons reduces the electrical activity firing rate, increases network synchronization, and for a sufficiently large HF stimulation, leads to complete electrical quiescence. In this study, we demonstrate a novel approach to investigate HF stimulation in biophysically-detailed ionic models of excitable cells, demonstrate possible mechanisms for HF stimulation conduction block in neurons, and provide insight into the influence of HF stimulation on neural networks.     

A neural network model based on the analogy with the immune system

The similarities between the immune system and the central nervous system lead to the formulation of an unorthodox neural network model. The similarities between the two systems are strong at the system level, but do not seem to be so striking at the level of the components. A new model of a neuron is therefore formulated, in order that the analogy can be used. The essential feature of the hypothetical neuron is that it exhibits hysteresis at the single neuron level. A network of N such neurons is modelled by an N-dimensional system of ordinary differential equations, which exhibits almost 2N attractors. The model has a property that resembles free will. A conjecture concerning how the network might learn stimulus-response behaviour is described. According to the conjecture, learning does not involve modifications of the strengths of synaptic connections. Instead, stimuli ("questions") selectively applied to the network by a "teacher" can be used to take the system to a region of the N-dimensional phase space where the network gives the desired stimulus-response behaviour. A key role for sleep in the learning process is suggested. The model for sleep leads to prediction that the variance in the rates of firing of the neurons associated with memory should increase during waking hours, and decrease during sleep.