This article has been reviewed according to Science X's editorial process and policies. Editors have highlighted the following attributes while ensuring the content's credibility: Physics is most readily applied to relatively simple systems: a pendulum, two electrons colliding or the structure of the solar system. But when systems become complicated—when many particles interact with one another, in condensed matter systems such as gases and fluids or in the cosmology of the early universe—simplifications can be made using a technique called classical or extended mean-field theory.
A research group from Princeton University, led by lead author Luca Di Carlo, has applied extended mean-field theory to networks of biological neurons. Their work is published in the journal Physical Review Letters. They found that the simplest versions of mean-field theory failed to accurately describe the activity of these networks, but an enhanced version was able to do so.
In a system with many particles, such as a two-dimensional lattice with particles at each vertex, each with a quantum spin (the two-dimensional Ising model), each particle interacts with all others in the lattice. That's almost always too unwieldy to obtain a complete solution for the model's properties, but an approximate solution can be found by assuming that each spin interacts with the average spin of the entire lattice, which is called classical mean-field theory. An extended mean-field theory would involve more information about the system's makeup, such as the probability distribution of neuron activity patterns.
In the latter case, fluctuations matter, not just the average of the system's variables. Neurons have an action potential—a rapid electrical pulse that travels along the neuron's axon, a long, slender projection that extends from the neuron's cell body. This action potential enables neurons, muscles and glands to communicate with one another.
This signal is binary—the cells send an electrical spike within a small time period, or they don't. So, at any given time, an assembly of neurons constitutes an Ising model. However, the group writes, "It proves surprisingly difficult to construct a consistent mean-field theory for patterns of activity in a network of real neurons." They continued, "We finally succeed with an extended model that matches the mean activity of individual neurons and the distribution of activity along one projection." At any given time, the network of neurons is in a particular configuration, with, in the language of Ising models, some "spin up" (with a spiked action potential) and others "spin down" (no action potential), with one binary possibility for each neuron.
The "state" of the neuronal network is the specification of the individual configurations of all the neurons. There are many possible states, and each configuration occurs with a certain probability. It's desirable to know how these possibilities are distributed, taking into account that the distribution must have an experimentally observed average spin and an observed correlation between any two spins.
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