IsingNeuronen

IsingNeuronen, or Ising neurons, are a simplified mathematical model for neural activity inspired by the Ising model from statistical physics. Each neuron is represented by a binary state, typically denoted s_i in {+1, -1}, where +1 corresponds to active and -1 to inactive. Neurons interact through symmetric couplings J_ij, which encode the strength and sign of pairwise connections, often with no self-coupling. An external field h_i can bias a neuron toward activation or silence.

The collective state of a network is described by an energy function E(s) = -1/2 sum_{i≠j} J_ij s_i s_j - sum_i h_i s_i. At finite temperature, the dynamics are stochastic: neurons flip states according to probabilistic rules (e.g., Glauber or Metropolis updates) that depend on the local field and temperature. At zero temperature, the system tends to minimize the energy, moving toward local or global minima.

IsingNeuronen are closely related to Hopfield networks, which are viewed as a practical implementation of symmetric Ising models for associative memory. Stored patterns correspond to energy minima, enabling the network to retrieve a complete pattern from partial or noisy cues. In neuroscience and machine learning, the Ising formulation is used to model population activity and to fit pairwise correlations in neural data via maximum entropy approaches.

Limitations include the binary, time-aggregated nature of states, lack of detailed biophysics, and the challenges of learning appropriate J_ij in biological systems. Extensions include kinetic or dynamic Ising models, asynchronous updates, and multi-state (Potts) variants that capture richer neural behavior.