Networks Of Excitable Elements Research Articles

Information processing in tree networks of excitable elements.

We study the collective response of small random tree networks of diffusively coupled excitable elements to stimuli applied to leaf nodes. Such networks model the morphology of certain sensory neurons that possess branched myelinated dendrites with excitable nodes of Ranvier at every branch point and at leaf nodes. Leaf nodes receive random inputs along with a stimulus and initiate action potentials that propagate through the tree. We quantify the collective response registered at the central node using mutual information. We show that in the strong-coupling limit, the statistics of the number of nodes and leaves determines the mutual information. At the same time, the collective response is insensitive to particular node connectivity and distribution of stimulus over leaf nodes. However, for intermediate coupling, the mutual information may strongly depend on the stimulus distribution among leaf nodes. We identify a mechanism behind the competition of leaf nodes that leads to nonmonotonous dependence of mutual information on coupling strength. We show that a localized stimulus given to a tree branch can be occluded by the background firing of unstimulated branches, thus suppressing mutual information. Nonetheless, the mutual information can be enhanced by a proper stimulus localization and tuning of coupling strength.

Backtracking activation impacts the criticality of excitable networks

Networks of excitable elements are widely used to model real-world biological and social systems. The dynamic range of an excitable network quantifies the range of stimulus intensities that can be robustly distinguished by the network response, and is maximized at the critical state. In this study, we examine the impacts of backtracking activation on system criticality in excitable networks consisting of both excitatory and inhibitory units. We find that, for dynamics with refractory states that prohibit backtracking activation, the critical state occurs when the largest eigenvalue of the weighted non-backtracking (WNB) matrix for excitatory units, , is close to one, regardless of the strength of inhibition. In contrast, for dynamics without refractory state in which backtracking activation is allowed, the strength of inhibition affects the critical condition through suppression of backtracking activation. As inhibitory strength increases, backtracking activation is gradually suppressed. Accordingly, the system shifts continuously along a continuum between two extreme regimes—from one where the criticality is determined by the largest eigenvalue of the weighted adjacency matrix for excitatory units, , to the other where the critical state is reached when is close to one. For systems in between, we find that and at the critical state. These findings, confirmed by numerical simulations using both random and synthetic neural networks, indicate that backtracking activation impacts the criticality of excitable networks.

Dynamic range maximization in excitable networks.

We study the strategy to optimally maximize the dynamic range of excitable networks by removing the minimal number of links. A network of excitable elements can distinguish a broad range of stimulus intensities and has its dynamic range maximized at criticality. In this study, we formulate the activation propagation in excitable networks as a message passing process in which a critical state is reached when the largest eigenvalue of the weighted non-backtracking matrix is close to one. By considering the impact of single link removal on the largest eigenvalue, we develop an efficient algorithm that aims to identify the optimal set of links whose removal will drive the system to the critical state. Comparisons with other competing heuristics on both synthetic and real-world networks indicate that the proposed method can maximize the dynamic range by removing the smallest number of links, and at the same time maintaining the largest size of the giant connected component.

Effect of small-world topology on wave propagation on networks of excitable elements

We study excitation waves on a Newman–Watts small-world network model of coupled excitable elements. Depending on the global coupling strength, we find differing resilience to the added long-range links and different mechanisms of propagation failure. For high coupling strengths, we show agreement between the network and a reaction-diffusion model with additional mean-field term. Employing this approximation, we are able to estimate the critical density of long-range links for propagation failure.

Synchronization in networks of excitable elements with time-delayed and rectifying coupling

Synchronization in networks of excitable elements with time-delayed and rectifying coupling

Dynamic patterns and their interactions in networks of excitable elements

Formation of localized propagating patterns is a fascinating self-organizing phenomenon that happens in a wide range of spatially extended, excitable systems in which individual elements have resting, activated, and refractory states. Here we study a type of stochastic three-state excitable network model that has been recently developed; this model is able to generate a rich range of pattern dynamics, including localized wandering patterns and localized propagating patterns with crescent shapes and long-range propagation. The collectivedynamicsof these localized patterns have anomalous subdiffusive dynamics before symmetry breaking and anomalous superdiffusive dynamics after that, showing long-range spatiotemporal coherence in the system. In this study, the stability of the localized wandering patterns is analyzed by treating an individual localized pattern as a subpopulation to develop its average response function. This stability analysis indicates that when the average refractory period is greater than a certain value, there are too many elements in the refractory state after being activated to allow the subpopulation to support a self-sustained pattern; this is consistent with symmetry breaking identified by using an order parameter. Furthermore, in a broad parameter space, the simple network model is able to generate a range of interactions between different localized propagating patterns including repulsive collisions and partial and full annihilations, and interactions between localized propagating patterns and the refractory wake behind others; in this study, these interaction dynamics are systematically quantified based on their relative propagation directions and the resultant angles between them before and after their collisions. These results suggest that the model potentially provides a modeling framework to understand the formation of localized propagating patterns in a broad class of systems with excitable properties.

Dynamic pattern formation and collisions in networks of excitable elements

Spatially extended, excitable systems with resting, activated, and refractory states, and emergent localized propagating patterns, are widespread in nature. Here a unique type of three-state excitable network model is shown to generate such dynamic patterns with rich collective dynamics. It is shown that symmetry breaking leads to the formation of dynamical patterns, leading to a change from local subdiffusive wandering to directed superdiffusive propagation. Furthermore, the model yields a rich repertoire of collision dynamics between localized propagating patterns and between propagating patterns and the refractory wakes of others. This work is particularly motivated by recent experimental studies of neural systems that exhibit localized propagating patterns, exemplifying a far wider class of excitable systems.

Input-output relation of FitzHugh-Nagumo elements arranged in a trifurcated structure

In this study, the propagation of an action potential in a network of excitable elements is studied numerically. The network we consider consists of excitable elements arranged in the shape of a trifurcated structure having three cables. The system has a branch point, a Y junction, at which the three cables are joined. Two types of external stimulations are considered: a single impulsive stimulation at one of the cable terminals, and a pair of stimuli applied to two different terminals. We have found three basic phases depending on the excitability of the elements for a single external stimulus as follows: (1) signal distributor--as the excitability gets higher, the pulse generated by a stimulus splits into two at the branch point, and two pulses are transmitted to the opposite terminals, (2) propagation block--the pulse in the lower excitable chain is blocked at the branch point, and (3) transient propagation--as the excitability is decreased further, we see that the pulse vanishes before reaching the branch point. By the interaction between the pulses that originate from different sources, signal transmission is recovered if the pulses arrive at the branch point nearly synchronously or after a specific delay time. The effects of the repetition of these two types of stimulation are also investigated. Complex spatiotemporal patterns occur due to pulse-pulse interaction and collisions at the branch point. The input-output relationship, which depends crucially on the repetition period and the time lag between the pair of stimuli, is characterized by the stimulus-response ratio and the interspike interval. We also show the effects of noise on the distribution of the interspike interval.

Physics of psychophysics: optimal dynamic range of critical excitable networks

A recurrent idea in the study of complex systems is that optimal information processing is to be found near phase transitions [1-3]. However, this heuristic hypothesis has few (if any) concrete realizations where a standard and biologically relevant quantity is optimized at criticality. Here we give a clear example of such phenomenon: a network of excitable elements has its sensitivity and dynamic range maximized at the critical point of a nonequilibrium phase transition. Our results are compatible with the essential role of gap junctions in olfactory glomeruli and retinal ganglion cell output. Synchronization and global oscillations also emerge from the network dynamics. We propose that the main functional role of electrical coupling is to provide an enhancement of dynamic range, therefore allowing the coding of information spanning several orders of magnitude. The mechanism provides a microscopic neural basis for psychophysical laws.

Optimal dynamical range of excitable networks at criticality

A recurrent idea in the study of complex systems is that optimal information processing is to be found near phase transitions. However, this heuristic hypothesis has few (if any) concrete realizations where a standard and biologically relevant quantity is optimized at criticality. Here we give a clear example of such a phenomenon: a network of excitable elements has its sensitivity and dynamic range maximized at the critical point of a non-equilibrium phase transition. Our results are compatible with the essential role of gap junctions in olfactory glomeruli and retinal ganglionar cell output. Synchronization and global oscillations also emerge from the network dynamics. We propose that the main functional role of electrical coupling is to provide an enhancement of dynamic range, therefore allowing the coding of information spanning several orders of magnitude. The mechanism could provide a microscopic neural basis for psychophysical laws.