We present generalized delayed neural network (DNN) model with positive delay feedback and neuron history. The local stability analysis around trivial local equilibria of delayed neural networks has applied and determine the conditions for the existence of zero root. We develop few innovative delayed neural network models in different dimensions through transformation and extension of some existing models. We found that zero root can have multiplicity two under certain conditions. We further show how the characteristic equation can have zero root and its multiplicity is dependent on the conditions undertaken. Finally, we generalize the neural network of $N$ neurons through which we determine the general form of Jacobian of the linear form and corresponding characteristic equation of the system.
We study the time delay in the synaptic conductance for suppression of spike synchronisation in a random network of Hodgkin Huxley neurons coupled by means of chemical synapses. In the first part, we examine in detail how the time delay acts over the network during the synchronised and desynchronised neuronal activities. We observe a relation between the neuronal dynamics and the syaptic conductance distributions. We find parameter values in which the time delay has high effectiveness in promoting the suppression of spike synchronisation. In the second part, we analyse how the delayed neuronal…
Mathematical modeling is an important tool to study the role of delay in neural systems and to evaluate its effects on the signaling activity of coupled neurons. Models for delayed neurons are often used to represent the dynamics of real neurons, but rarely to assess the energy required to maintain these dynamics. In this work, we address these questions from an energy perspective by considering a pair of Hindmarsh-Rose burst neurons coupled by reciprocal time-delayed coupling with electrical and chemical synapses. We examine the average energy consumption required to maintain cooperative beha…
Collective chaos is shown to emerge, via a period-doubling cascade, from quasiperiodic partial synchronization in a population of identical inhibitory neurons with delayed global coupling. This system is thoroughly investigated by means of an exact model of the macroscopic dynamics, valid in the thermodynamic limit. The collective chaotic state is reproduced numerically with a finite population, and persists in the presence of weak heterogeneities. Finally, the relationship of the model's dynamics with fast neuronal oscillations is discussed.
For a class of fast {\it Cl-}type inhibitory spiking neuron models with delayed feedback fed with a Poisson stochastic process of excitatory impulses, it is proven that the stream of output interspike intervals cannot be presented as a Markov process of any order.
We consider a wide class of spiking neuron models, defined by rather general set of conditions typical for basic models like leaky integrate and fire, or binding neuron model. A neuron is fed with a point renewal process. A relation between the three probability density functions (pdf): (i) pdf of input interspike intervals, (ii) pdf of output interspike intervals of a neuron with instantaneous feedback and (iii) pdf for that same neuron without feedback is derived. This allows to calculate any of the three pdfs provided the another two are given. Similar relation between corresponding means a…
A binding neuron (BN) whith delayed feedback is considered. The neuron is fed externally with a Poisson stream of intensity $λ$. The neuron's output spikes are fed into its input with time delay $Δ$. The resulting output stream of the BN is not Poissonian, and we look for its interspike intervals (ISI) distribution. For BN with threshold 2 an exact mathematical expression as function of $λ$, $Δ$ and BN's internal memory, $τ$ is derived for the ISI distribution, and for higher thresholds it is found numerically. The distributions found are characterized with discontinuities of jump type, and in…
For a class of excitatory spiking neuron models with delayed feedback fed with a Poisson stochastic process, it is proven that the stream of output interspike intervals cannot be presented as a Markov process of any order. Keywords: spiking neuron; Poisson stochastic process; probability density function; delayed feedback; non-Markov stochastic process
Neurons are connected to other neurons by axons and dendrites that conduct signals with finite velocities, resulting in delays between the firing of a neuron and the arrival of the resultant impulse at other neurons. Since delays greatly complicate the analytical treatment and interpretation of models, they are usually neglected or taken to be uniform, leading to a lack in the comprehension of the effects of delays in neural systems. This paper shows that heterogeneous transmission delays make small groups of neurons respond selectively to inputs with differing frequency spectra. By studying a…
We consider a class of spiking neuronal models, defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire or the binding neuron model and also for some artificial neurons. A neuron is fed with a Poisson process. Each output impulse is applied to the neuron itself after a finite delay $Δ$. This impulse acts as being delivered through a fast Cl-type inhibitory synapse. We derive a general relation which allows calculating exactly the probability density function (pdf) $p(t)$ of output interspike intervals of a neuron with feedback based on known…
Here we investigate the synchronization of networks of FitzHugh-Nagumo neurons coupled in scale-free, small-world and random topologies, in the presence of distributed time delays in the coupling of neurons. We explore how the synchronization transition is affected when the time delays in the interactions between pairs of interacting neurons are non-uniform. We find that the presence of distributed time-delays does not change the behavior of the synchronization transition significantly, vis-a-vis networks with constant time-delay, where the value of the constant time-delay is the mean of the d…
A class of spiking neuronal models with threshold 2 is considered. It is defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire (LIF) or the binding neuron model and also for some artificial neurons. A neuron is stimulated with a Poisson stream of excitatory impulses. Each output impulse is conveyed through the feedback line to the neuron input after finite delay $Δ$. This impulse is identical to those delivered from the input stream. We have obtained a general relation allowing calculating exactly the probability density function (PDF) $p(…
Sensory neurons are often described in terms of a receptive field, that is, a linear kernel through which stimuli are filtered before they are further processed. If information transmission is assumed to proceed in a feedforward cascade, the receptive field may be interpreted as the external stimulus' profile maximizing neuronal output. The nervous system, however, contains many feedback loops, and sensory neurons filter more currents than the ones representing the transduced external stimulus. Some of the additional currents are generated by the output activity of the neuron itself, and there…
We analyze a large system of heterogeneous quadratic integrate-and-fire (QIF) neurons with time delayed, all-to-all synaptic coupling. The model is exactly reduced to a system of firing rate equations that is exploited to investigate the existence, stability and bifurcations of fully synchronous, partially synchronous, and incoherent states. In conjunction with this analysis we perform extensive numerical simulations of the original network of QIF neurons, and determine the relation between the macroscopic and microscopic states for partially synchronous states. The results are summarized in t…
We study synaptically coupled neuronal networks to identify the role of coupling delays in network's synchronized behaviors. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled and interact with each other. The excitatory population is uncoupled, while the inhibitory population is tightly coupled. A geometric singular perturbation analysis yields existence and stability conditions for synchronization states under different firing patterns between the two populations, along with formulas for the periods…
Several studies with brain signals suggested that bottom-up and top-down influences are exerted through distinct frequency bands among visual cortical areas. It has been recently shown that theta and gamma rhythms subserve feedforward, whereas the feedback influence is dominated by the alpha-beta rhythm in primates. A few theoretical models for reproducing these effects have been proposed so far. Here we show that a simple but biophysically plausible two-network motif composed of spiking-neuron models and chemical synapses can exhibit feedforward and feedback influences through distinct freque…
Message passing between components of a distributed physical system is non-instantaneous and contributes to determine the time scales of the emerging collective dynamics like an effective inertia. In biological neuron networks this inertia is due in part to local synaptic filtering of exchanged spikes, and in part to the distribution of the axonal transmission delays. How differently these two kinds of inertia affect the network dynamics is an open issue not yet addressed due to the difficulties in dealing with the non-Markovian nature of synaptic transmission. Here, we develop a mean-field di…
Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show th…
Hysteresis and multistability in network spreading with neuronal activity feedback
Wu Tsai Institute, Yale University（耶鲁大学吴-tsai研究所）；Department of Psychology, Yale University（耶鲁大学心理学系）；Department of Biomedical Engineering, Yale University（耶鲁大学生物医学工程系）
Brain plasticity refers to brain's ability to change neuronal connections, as a result of environmental stimuli, new experiences, or damage. In this work, we study the effects of the synaptic delay on both the coupling strengths and synchronisation in a neuronal network with synaptic plasticity. We build a network of Hodgkin-Huxley neurons, where the plasticity is given by the Hebbian rules. We verify that without time delay the excitatory synapses became stronger from the high frequency to low frequency neurons and the inhibitory synapses increases in the opposite way, when the delay is incre…
We study a network of spiking neurons with heterogeneous excitabilities connected via inhibitory delayed pulses. For globally coupled systems the increase of the inhibitory coupling reduces the number of firing neurons by following a Winner Takes All mechanism. For sufficiently large transmission delay we observe the emergence of collective oscillations in the system beyond a critical coupling value. Heterogeneity promotes neural inactivation and asynchronous dynamics and its effect can be counteracted by considering longer time delays. In sparse networks, inhibition has the counterintuitive e…
Synchronized bursts (SBs) with complex structures are common in neuronal cultures. Although the origin of SBs is still unclear, they have been studied for their information processing capabilities. Here, we investigate the properties of these SBs in a culture on multi-electrode array system. We find that structures of these SBs are related to the different developmental stages of the cultures. A model based on short term synaptic plasticity, recurrent connections and astrocytic recycling of neurotransmitters has been developed successfully to understand these structures. A phase diagram obtain…
We consider a simple Markovian class of the stochastic Wilson-Cowan type models of neuronal network dynamics, which incorporates stochastic delay caused by the existence of a refractory period of neurons. From the point of view of the dynamics of the individual elements, we are dealing with a network of non-Markovian stochastic two-state oscillators with memory which are coupled globally in a mean-field fashion. This interrelation of a higher-dimensional Markovian and lower-dimensional non-Markovian dynamics is discussed in its relevance to the general problem of the network dynamics of comple…
We address the problem of learning feedback control where the controller is a network constructed solely of deterministic spiking neurons. In contrast to previous investigations that were based on a spike rate model of the neuron, the control signal here is determined by the precise temporal positions of spikes generated by the output neurons of the network. We model the problem formally as a hybrid dynamical system comprised of a closed loop between a plant and a spiking neuron network. We derive a novel synaptic weight update rule via which the spiking neuron network controller learns to hol…
The oculomotor delayed-response (ODR) task is a common experimental paradigm of working memory (WM) study, in which a monkey must fixate its gaze on the center of a screen and, following a brief cue that flashes on the screen, keep fixating for several more seconds before shifting its gaze to the location where the cue flashed. Consequently, in the delay period between the cue and the response the monkey must maintain a memory of cue location. Electrophysiological recordings from the prefrontal area of the cortex (PFC) revealed neurons that display selective persistent activity: their firing r…
Delay coordinates synchronization and induces abrupt transition in excitable networks
Institute of Science and Technology, Universidade Federal de São Paulo (UNIFESP)（圣保罗联邦大学科学与技术学院）；Cellular Computations and Learning, Max Planck Institute for Neurobiology of Behavior（马克斯·普朗克行为神经生物学研究所细胞计算与学习部门）；Department of Mathematics, Western University（韦仕敦大学数学系）；Fields Lab for Network Computation, Fields Institute（菲尔兹研究所网络计算实验室）；Department of Neuroscience, University of Lethbridge（莱斯布里奇大学生物神经科学系）
The Unitary Events (UE) method is a popular and efficient method used this last decade to detect dependence patterns of joint spike activity among simultaneously recorded neurons. The first introduced method is based on binned coincidence count \citep{Grun1996} and can be applied on two or more simultaneously recorded neurons. Among the improvements of the methods, a transposition to the continuous framework has recently been proposed in \citep{muino2014frequent} and fully investigated in \citep{MTGAUE} for two neurons. The goal of the present paper is to extend this study to more than two neu…
Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we analyze the mesoscopic and macroscopic limits of networks with random correlated connectivity weights and delays. We address both averaged and quenched limits, and show propagation of chaos and convergence to a complex integral McKean-Vlasov equations with distributed delays. We then instantiate a compl…
Noise-delayed decay (NDD) phenomenon emerges when the first-spike latency of a periodically forced stochastic neuron exhibits a maximum for a particular range of noise intensity. Here, we investigate the latency response dynamics of a single Hodgkin-Huxley neuron that is subject to both a suprathreshold periodic stimulus and a background activity arriving through dynamic synapses. We study the first spike latency response as a function of the presynaptic firing rate f. This constitutes a more realistic scenario than previous works, since f provides a suitable biophysically realistic parameter …
Critical brain hypothesis has been intensively studied both in experimental and theoretical neuroscience over the past two decades. However, some important questions still remain: (i) What is the critical point the brain operates at? (ii) What is the regulatory mechanism that brings about and maintains such a critical state? (iii) The critical state is characterized by scale-invariant behavior which is seemingly at odds with definitive brain oscillations? In this work we consider a biologically motivated model of Izhikevich neuronal network with chemical synapses interacting via spike-timingde…
The plasticity of the conduction delay between neurons plays a fundamental role in learning. However, the exact underlying mechanisms in the brain for this modulation is still an open problem. Understanding the precise adjustment of synaptic delays could help us in developing effective brain-inspired computational models in providing aligned insights with the experimental evidence. In this paper, we propose an unsupervised biologically plausible learning rule for adjusting the synaptic delays in spiking neural networks. Then, we provided some mathematical proofs to show that our learning rule …
Analysing correlations between streams of events is an important problem. It arises for example in Neurosciences, when the connectivity of neurons should be inferred from spike trains that record neurons' individual spiking activity. While recently some approaches for inferring delayed synaptic connections have been proposed, they are limited in the types of connectivities and delays they are able to handle, or require computation-intensive procedures. This paper proposes a faster and more flexible approach for analysing such delayed correlated activity: a statistical approach for the Analysis…
