Hopfield networkA Hopfield network (or Amari-Hopfield network, Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described by Shun'ichi Amari in 1972 and by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz on the Ising model. Hopfield networks serve as content-addressable ("associative") memory systems with binary threshold nodes, or with continuous variables.
Social dynamicsSocial dynamics (or sociodynamics) is the study of the behavior of groups that results from the interactions of individual group members as well to the study of the relationship between individual interactions and group level behaviors. The field of social dynamics brings together ideas from economics, sociology, social psychology, and other disciplines, and is a sub-field of complex adaptive systems or complexity science. The fundamental assumption of the field is that individuals are influenced by one another's behavior.
Scale-free networkA scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as where is a parameter whose value is typically in the range (wherein the second moment (scale parameter) of is infinite but the first moment is finite), although occasionally it may lie outside these bounds. The name "scale-free" means that some moments of the degree distribution are not defined, so that the network does not have a characteristic scale or "size".
Logistic mapThe logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written where xn is a number between zero and one, which represents the ratio of existing population to the maximum possible population.
Network dynamicsNetwork dynamics is a research field for the study of networks whose status changes in time. The dynamics may refer to the structure of connections of the units of a network, to the collective internal state of the network, or both. The networked systems could be from the fields of biology, chemistry, physics, sociology, economics, computer science, etc. Networked systems are typically characterized as complex systems consisting of many units coupled by specific, potentially changing, interaction topologies.
Small-world networkA small-world network is a mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other. Due to this, most neighboring nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is: while the global clustering coefficient is not small.
Reservoir computingReservoir computing is a framework for computation derived from recurrent neural network theory that maps input signals into higher dimensional computational spaces through the dynamics of a fixed, non-linear system called a reservoir. After the input signal is fed into the reservoir, which is treated as a "black box," a simple readout mechanism is trained to read the state of the reservoir and map it to the desired output. The first key benefit of this framework is that training is performed only at the readout stage, as the reservoir dynamics are fixed.
Spike-and-waveSpike-and-wave is a pattern of the electroencephalogram (EEG) typically observed during epileptic seizures. A spike-and-wave discharge is a regular, symmetrical, generalized EEG pattern seen particularly during absence epilepsy, also known as ‘petit mal’ epilepsy. The basic mechanisms underlying these patterns are complex and involve part of the cerebral cortex, the thalamocortical network, and intrinsic neuronal mechanisms. The first spike-and-wave pattern was recorded in the early twentieth century by Hans Berger.
Network theoryIn mathematics, computer science and network science, network theory is a part of graph theory. It defines networks as graphs where the nodes or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components. Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering, biology, archaeology, linguistics, economics, finance, operations research, climatology, ecology, public health, sociology, psychology, and neuroscience.
Dynamical systems theoryDynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.
