On the Phase-Space Dynamics of Systems of Spiking Neurons. II: Formal Analysis (original) (raw)
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On the Phase-Space Dynamics of Systems of Spiking Neurons. I: Model and Experiments
2000
We begin with a brief review of the abstract dynamical system that models systems of biological neurons, introduced in the original article. We then analyze the dynamics of the system. Formal analysis of local properties of ows reveals contraction, expansion, and folding in different sections of the phase-space. The criterion for the system, set up to model a typical neocortical column, to be sensitive to initial conditions is identi ed. Based on physiological param eters, we then deduce that periodic orbits in the region of the phase-space corresponding to normal operational conditions in the neocortex are almost surely (with probability 1) unstable, those in the region corresponding to seizure-like conditions are almost surely stable, and trajectories in the region corresponding to normal operational conditions are almost surely sensitive to initial conditions. Next, we present a procedure that isolates all basic sets, complex sets, and attractors incrementally. Based on the two sets of results, we conclude that chaotic attractors that are potentially anisotropic play a central role in the dynamics of such systems. Finally, we examine the impact of this result on the computational nature of neocortical neuronal systems.
2015
We investigate the phase-space dynamics of a general model of local sys-tems of biological neurons in order to deduce the salient dynamical char-acteristics of such systems. In this article, we present a detailed exposition of an abstract dynamical system that models systems of biological neu-rons. The abstract system is based on a limited set of realistic assumptions and thus accommodates a wide range of neuronal models. Simulation re-sults are presented for several instantiations of the abstract system, each modeling a typical neocortical column to a different degree of accuracy. The results demonstrate that the dynamics of the systems are generally consistent with that observed in neurophysiological experiments. They reveal that the qualitative behavior of the class of systems can be clas-sied into three distinct categories: quiescence, intense periodic activity resembling a state of seizure, and sustained chaos over the range of in-trinsic activity typically associated with norm...
Chaotic oscillations in a map-based model of neural activity
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2007
We propose a discrete time dynamical system (a map) as phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find condition under which this map has an invariant region on the phase plane, containing chaotic attractor.
Spiking dynamics of interacting oscillatory neurons
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2005
Spiking sequences emerging from dynamical interaction in a pair of electronic neurons is investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh-Nagumo units with modified excitability (MFHN). The The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. The spiking phase maps describing the response dynamics are investigated. Complex phase locking and chaotic sequences are analyzed. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the possibility of neuronal information encoding.
Theory of Coupled Neuronal-Synaptic Dynamics
2024
In neural circuits, synaptic strengths influence neuronal activity by shaping network dynamics, and neuronal activity influences synaptic strengths through activity-dependent plasticity. Motivated by this fact, we study a recurrent-network model in which neuronal units and synaptic couplings are interacting dynamic variables, with couplings subject to Hebbian modification with decay around quenched random strengths. Rather than assigning a specific role to the plasticity, we use dynamical mean-field theory and other techniques to systematically characterize the neuronal-synaptic dynamics, revealing a rich phase diagram. Adding Hebbian plasticity slows activity in already chaotic networks and can induce chaos in otherwise quiescent networks. Anti-Hebbian plasticity quickens activity and produces an oscillatory component. Analysis of the Jacobian shows that Hebbian and anti-Hebbian plasticity push locally unstable modes toward the real and imaginary axes, respectively, explaining these behaviors. Both random-matrix and Lyapunov analysis show that strong Hebbian plasticity segregates network timescales into two bands, with a slow, synapse-dominated band driving the dynamics, suggesting a flipped view of the network as synapses connected by neurons. For increasing strength, Hebbian plasticity initially raises the complexity of the dynamics, measured by the maximum Lyapunov exponent and attractor dimension, but then decreases these metrics, likely due to the proliferation of stable fixed points. We compute the marginally stable spectra of such fixed points as well as their number, showing exponential growth with network size. Finally, in chaotic states with strong Hebbian plasticity, a stable fixed point of neuronal dynamics is destabilized by synaptic dynamics, allowing any neuronal state to be stored as a stable fixed point by halting the plasticity. This phase of freezable chaos offers a new mechanism for working memory.
Phase description of spiking neuron networks with global electric and synaptic coupling
Physical Review E, 2011
Phase models are among the simplest neuron models reproducing spiking behavior, excitability, and bifurcations toward periodic firing. However, coupling among neurons has been considered only using generic arguments valid close to the bifurcation point, and the differentiation between electric and synaptic coupling remains an open question. In this work we aim to address this question and derive a mathematical formulation for the various forms of coupling. We construct a mathematical model based on a planar simplification of the Morris-Lecar model. Based on geometric arguments we then derive a phase description of a network of the above oscillators with biologically realistic electric coupling and subsequently with chemical coupling under fast synapse approximation. We demonstrate analytically that electric and synaptic coupling are differently expressed on the level of the network's phase description with qualitatively different dynamics. Our mathematical analysis shows that a breaking of the translational symmetry in the phase flows is responsible for the different bifurcations paths of electric and synaptic coupling. Our numerical investigations confirm these findings and show excellent correspondence between the dynamics of the full network and the network's phase description.
Dynamical behavior of the firings in a coupled neuronal system
Physical Review E, 1993
The time-interval sequences and the spatiotemporal patterns of the firings of a coupled neuronal network are investigated in this paper. For a single neuron stimulated by an external stimulus I, the timeinterval sequences show a low-frequency firing of bursts of spikes and a reversed period-doubling cascade to a high-frequency repetitive firing state as the stimulus Iis increased. For two neurons coupled to each other through the firing of the spikes, the complexity of the time-interval sequences becomes simple as the coupling strength increases. A network with a large number of neurons shows a complex spatiotemporal pattern structure. As the coupling strength increases, the number of phase-locked neurons increases and the time-interval diagram shows temporal chaos and a bifurcation in the space. The dynamical behavior is also verified by the behavior of the Lyapunov exponent.
Oscillatory activity in excitable neural systems
Contemporary Physics, 2010
The brain is a complex system and exhibits various subsystems on different spatial and temporal scales. These subsystems are recurrent networks of neurons or populations that interact with each other. The single neurons are microscopic objects and evolve on a different time scale than macroscopic neural populations. To understand the dynamics of the brain, however, it is necessary to understand the dynamics of the brain network both on the microscopic and the macroscopic level and the interaction between the levels. The presented work introduces to the major properties of single neurons and their interactions. The physical aspects of some standard mathematical models are discussed in some detail. The work shows that both single neurons and neural populations are excitable in the sense that small differences in an initial short stimulation may yield very different dynmical behavior of the system. To illustrate the power of the neural population model discussed, the work applies the model to explain experimental activity in the delayed feedback system in weakly electric fish and the electroencephalogram (EEG).
Frontiers in Systems Neuroscience, 2014
Complex collective activity emerges spontaneously in cortical circuits in vivo and in vitro, such as alternation of up and down states, precise spatiotemporal patterns replay, and power law scaling of neural avalanches. We focus on such critical features observed in cortical slices. We study spontaneous dynamics emerging in noisy recurrent networks of spiking neurons with sparse structured connectivity. The emerging spontaneous dynamics is studied, in presence of noise, with fixed connections. Note that no short-term synaptic depression is used. Two different regimes of spontaneous activity emerge changing the connection strength or noise intensity: a low activity regime, characterized by a nearly exponential distribution of firing rates with a maximum at rate zero, and a high activity regime, characterized by a nearly Gaussian distribution peaked at a high rate for high activity, with long-lasting replay of stored patterns. Between this two regimes, a transition region is observed, where firing rates show a bimodal distribution, with alternation of up and down states. In this region, one observes neuronal avalanches exhibiting power laws in size and duration, and a waiting time distribution between successive avalanches which shows a non-monotonic behavior. During periods of high activity (up states) consecutive avalanches are correlated, since they are part of a short transient replay initiated by noise focusing, and waiting times show a power law distribution. One can think at this critical dynamics as a reservoire of dynamical patterns for memory functions. Citation: Scarpetta S and de Candia A (2014) Alternation of up and down states at a dynamical phase-transition of a neural network with spatiotemporal attractors. Front. Syst. Neurosci. 8:88.