Networks Of Excitable Neurons Research Articles

Evolution of neuroglia.

The evolution of the nervous system progressed through cellular diversification and specialization of functions. Conceptually, the nervous system is composed of electrically excitable neuronal networks connected by chemical synapses and nonexcitable glial cells that provide for homeostasis and defense. The evolution of neuroglia began with the emergence of the centralized nervous system and proceeded through a continuous increase in their complexity. In the primate brain, especially in the brain of humans, the astrocyte lineage is exceedingly complex, with the emergence of new types of astroglial cells possibly involved in interlayer communication and integration.

Influence of inhibitory synapses on the criticality of excitable neuronal networks

In this work, we study the dynamic range of a neuronal network of excitable neurons with excitatory and inhibitory synapses. We obtain an analytical expression for the critical point as a function of the excitatory and inhibitory synaptic intensities. We also determine an analytical expression that gives the critical point value in which the maximal dynamic range occurs. Depending on the mean connection degree and coupl\-ing weights, the critical points can exhibit ceasing or ceaseless dynamics. However, the dynamic range is equal in both cases. We observe that the external stimulus mask some effects of self-sustained activity (ceaseless dynamic) in the region where the dynamic range is calculated. In these regions, the firing rate is the same for ceaseless dynamics and ceasing activity. Furthermore, we verify that excitatory and inhibitory inputs are approximately equal for a network with a large number of connections, showing excitatory-inhibitory balance as reported experimentally.

Quantum information theoretic approach to the mind–brain problem

The brain is composed of electrically excitable neuronal networks regulated by the activity of voltage-gated ion channels. Further portraying the molecular composition of the brain, however, will not reveal anything remotely reminiscent of a feeling, a sensation or a conscious experience. In classical physics, addressing the mind–brain problem is a formidable task because no physical mechanism is able to explain how the brain generates the unobservable, inner psychological world of conscious experiences and how in turn those conscious experiences steer the underlying brain processes toward desired behavior. Yet, this setback does not establish that consciousness is non-physical. Modern quantum physics affirms the interplay between two types of physical entities in Hilbert space: unobservable quantum states, which are vectors describing what exists in the physical world, and quantum observables, which are operators describing what can be observed in quantum measurements. Quantum no-go theorems further provide a framework for studying quantum brain dynamics, which has to be governed by a physically admissible Hamiltonian. Comprising consciousness of unobservable quantum information integrated in quantum brain states explains the origin of the inner privacy of conscious experiences and revisits the dynamic timescale of conscious processes to picosecond conformational transitions of neural biomolecules. The observable brain is then an objective construction created from classical bits of information, which are bound by Holevo’s theorem, and obtained through the measurement of quantum brain observables. Thus, quantum information theory clarifies the distinction between the unobservable mind and the observable brain, and supports a solid physical foundation for consciousness research.

Drifting States and Synchronization Induced Chaos in Autonomous Networks of Excitable Neurons.

The study of balanced networks of excitatory and inhibitory neurons has led to several open questions. On the one hand it is yet unclear whether the asynchronous state observed in the brain is autonomously generated, or if it results from the interplay between external drivings and internal dynamics. It is also not known, which kind of network variabilities will lead to irregular spiking and which to synchronous firing states. Here we show how isolated networks of purely excitatory neurons generically show asynchronous firing whenever a minimal level of structural variability is present together with a refractory period. Our autonomous networks are composed of excitable units, in the form of leaky integrators spiking only in response to driving currents, remaining otherwise quiet. For a non-uniform network, composed exclusively of excitatory neurons, we find a rich repertoire of self-induced dynamical states. We show in particular that asynchronous drifting states may be stabilized in purely excitatory networks whenever a refractory period is present. Other states found are either fully synchronized or mixed, containing both drifting and synchronized components. The individual neurons considered are excitable and hence do not dispose of intrinsic natural firing frequencies. An effective network-wide distribution of natural frequencies is however generated autonomously through self-consistent feedback loops. The asynchronous drifting state is, additionally, amenable to an analytic solution. We find two types of asynchronous activity, with the individual neurons spiking regularly in the pure drifting state, albeit with a continuous distribution of firing frequencies. The activity of the drifting component, however, becomes irregular in the mixed state, due to the periodic driving of the synchronized component. We propose a new tool for the study of chaos in spiking neural networks, which consists of an analysis of the time series of pairs of consecutive interspike intervals. In this space, we show that a strange attractor with a fractal dimension of about 1.8 is formed in the mentioned mixed state.

The homeostatic astroglia emerges from evolutionary specialization of neural cells.

Evolution of the nervous system progressed through cellular diversification and specialization of functions. Conceptually, the nervous system is composed from electrically excitable neuronal networks connected with chemical synapses and non-excitable glial cells that provide for homeostasis and defence. Astrocytes are integrated into neural networks through multipartite synapses; astroglial perisynaptic processes closely enwrap synaptic contacts and control homeostasis of the synaptic cleft, supply neurons with glutamate and GABA obligatory precursor glutamine and contribute to synaptic plasticity, learning and memory. In neuropathology, astrocytes may undergo reactive remodelling or degeneration; to a large extent, astroglial reactions define progression of the pathology and neurological outcome.This article is part of the themed issue 'Evolution brings Ca(2+) and ATP together to control life and death'.

Spike coherence and synchronization on Newman–Watts small-world neuronal networks modulated by spike-timing-dependent plasticity

The effects of spike-timing-dependent plasticity (STDP) and noise intensity on the temporal and spatial dynamics of Newman–Watts small-world neuronal networks are studied. Numerical results show that, an intermediate intensity of additive noise can optimize the dynamical response of the neural system, where the noise-induced coherence resonance and spiking synchronization occur. The adaptive coupling modulated by STDP can largely depress the temporal coherence and spatial synchrony induced by external noise and random shortcuts. In particular, as the adjusting rate increases, lower noise intensity is needed to maximize the networked synchronization, and more connections are introduced to achieve coherence resonance. Moreover, the small-world topology can significantly affect the dynamics of excitable neuronal networks. It is found that the temporal coherence of neuronal activity reaches peaks for an appropriate number of random shortcuts, while the spiking synchronization is always enhanced as more shortcuts are added into the network.

Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks.

The phenomenon of stochastic resonance in Newman-Watts small-world neuronal networks is investigated when the strength of synaptic connections between neurons is adaptively adjusted by spike-time-dependent plasticity (STDP). It is shown that irrespective of the synaptic connectivity is fixed or adaptive, the phenomenon of stochastic resonance occurs. The efficiency of network stochastic resonance can be largely enhanced by STDP in the coupling process. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed one when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, we demonstrate that the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of the adaptive network. Furthermore, the small-world topology can significantly affect stochastic resonance of excitable neuronal networks. It is found that there exists an optimal probability of adding links by which the noise-induced transmission of weak periodic signal peaks.

Effects of time delay and random rewiring on the stochastic resonance in excitable small-world neuronal networks

The effects of time delay and rewiring probability on stochastic resonance and spatiotemporal order in small-world neuronal networks are studied in this paper. Numerical results show that, irrespective of the pacemaker introduced to one single neuron or all neurons of the network, the phenomenon of stochastic resonance occurs. The time delay in the coupling process can either enhance or destroy stochastic resonance on small-world neuronal networks. In particular, appropriately tuned delays can induce multiple stochastic resonances, which appear intermittently at integer multiples of the oscillation period of the pacemaker. More importantly, it is found that the small-world topology can significantly affect the stochastic resonance on excitable neuronal networks. For small time delays, increasing the rewiring probability can largely enhance the efficiency of pacemaker-driven stochastic resonance. We argue that the time delay and the rewiring probability both play a key role in determining the ability of the small-world neuronal network to improve the noise-induced outreach of the localized subthreshold pacemaker.

Phase-disorder-induced firing activity in excitable neuronal networks with attractive and repulsive coupling

It has been revealed that the network of excitable neurons via attractive coupling can generate spikes under stimuli of subthreshold signals with disordered phases. In this paper, we explore the firing activity induced by phase disorder in excitable neuronal networks consisting of both attractive and repulsive coupling. By increasing the fraction of repulsive coupling, we find that, in the weak coupling strength case, the firing threshold of phase disorder is increased and the system response to subthreshold signals is decreased, indicating that the effect of inducing neuron firing by phase disorder is weakened with repulsive coupling. Interestingly, in the large coupling strength case, we see an opposite situation, where the coupled neurons show a rather large response to the subthreshold signals even with small phase disorder. The latter case implies that the effect of phase disorder is enhanced by repulsive coupling. A system of two-coupled excitable neurons is used to explain the role of repulsive coupling on phase-disorder-induced firing activity.

The Geometry of Spontaneous Spiking in Neuronal Networks

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze the main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we describe the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulas for the network activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior.

Splay States in Finite Pulse-Coupled Networks of Excitable Neurons

The emergence and stability of splay states is studied in fully coupled finite networks of $N$ excitable quadratic integrate-and-fire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (pulse emission and termination), we were able to write down an exact event-driven map for the system and to evaluate the splay state solutions. For $M$ overlapping postsynaptic potentials, the linear stability analysis of the splay state should also take in account, besides the actual values of the membrane potentials, the firing times associated with the $M$ previous pulse emissions. As a matter of fact, it was possible, by introducing $M$ complementary variables, to rephrase the evolution of the network as an event-driven map and to derive an analytic expression for the Floquet spectrum. We find that, independently of $M$, the splay state is marginally stable with $N-2$ neutral directions. Furthermore, we have identified a family of periodic solutions surrounding the splay state and sharing the same neutral stability directions. In the limit of $\delta$-pulses, it is still possible to derive an event-driven formulation for the dynamics; however, the number of neutrally stable directions associated with the splay state becomes $N$. Finally, we prove a link between the results for our system and a previous theory [S. Watanabe and S. H. Strogatz, Phys. D, 74 (1994), pp. 197--253] developed for networks of phase oscillators with sinusoidal coupling.

Synchronization and spontaneous dynamics in the Locus Coeruleus network

The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented in this work. Using the Freidlin-Wentzell large deviation theory for randomly perturbed dynamical systems and the elements of the algebraic graph theory, we identify and analyze main regimes in the network dynamics in terms of the key control parameters: excitability, coupling strength, and network topology. The analysis reveals the geometry of spontaneous dynamics in electrically coupled network. Specifically, we show that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network. By studying how the minima of this function evolve under the variation of the coupling strength, we identify the principal transformations in the network dynamics. The minimization problem is also used for the quantitative description of the main dynamical regimes and transitions between them. In particular, for the weak and strong coupling regimes, we present asymptotic formulae for the activity rate as a function of the coupling strength and the degree of the network. The variational analysis is complemented by the stability analysis of the synchronous state in the strong coupling regime. The stability estimates reveal the contribution of the network connectivity and the properties of the cycle subspace associated with the graph of the network to its synchronization properties. This work is motivated by the experimental and modeling studies of the ensemble of neurons in the Locus Coeruleus, a nucleus in the brainstem involved in the regulation of cognitive performance and behavior [1].

Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons

We study the dynamical states of a small-world network of recurrently coupled excitable neurons, through both numerical and analytical methods. The dynamics of this system depend mostly on both the number of long-range connections or "shortcuts", and the delay associated with neuronal interactions. We find that persistent activity emerges at low density of shortcuts, and that the system undergoes a transition to failure as their density reaches a critical value. The state of persistent activity below this transition consists of multiple stable periodic attractors, whose number increases at least as fast as the number of neurons in the network. At large shortcut density and for long enough delays the network dynamics exhibit exceedingly long chaotic transients, whose failure times follow a stretched exponential distribution. We show that this functional form arises for the ensemble-averaged activity if the failure time for each individual network realization is exponentially distributed.

Calcium ions and integration in neural circuits

Integration in the nervous system is achieved by signal processing within dynamic functional ensembles formed by highly complex neuronal-glial cellular circuits. The interactions between electrically excitable neuronal networks and electrically non-excitable glial syncytium occur through either chemical transmission, which involves the release of transmitters from presynaptic terminals or from astroglial cells, or via direct intercellular contacts, gap junctions. Calcium ions act as a universal intracellular signalling system, which controls many aspects of neuronal-glial communications. In neurones, calcium signalling events regulate the exocytosis of neurotransmitters and establish the link between excitation of postsynaptic cells and integrative intracellular events, which control synaptic strength, expression of genes and memory function. In glial cells metabotropic receptor mediated release of calcium ions from the intracellular endoplasmic reticulum calcium store provide specific form of glial excitability. Glial calcium signals ultimately result in vesicular secretion of "glio" transmitters, which affect neuronal networks thus closing the glial-neuronal circuits. Cellular signalling through calcium ions therefore can be regarded as a molecular mechanism of integration in the nervous system.

Synchronization of pulse-coupled excitable neurons.

Collective behavior of pulse-coupled oscillatory neurons has been investigated widely. In many cases, however, real neurons are intrinsically not oscillatory but excitable. The networks of excitable neurons can have their own characteristic dynamics, and they are of interest also from the viewpoint of functional assemblies. In the present paper, the collective behavior of pulse-coupled excitable neurons has been investigated using phase description. It is shown that full synchronization is achieved in networks of excitable leaky integrate-and-fire neurons and discrete-time Nagumo-Sato neurons. The cooperative roles of external spike inputs, decay of internal states, and feedback spikes are explained. Enhancement of synchronization by refractoriness and noise is also reported.