Statistical mechanics for natural flocks of birds

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Abstract

Interactions among neighboring birds in a flock cause an alignment of their flight directions. We show that the minimally structured (maximum entropy) model consistent with these local correlations correctly predicts the propagation of order throughout entire flocks of starlings, with no free parameters. These models are mathematically equivalent to the Heisenberg model of magnetism, and define an "energy" for each configuration of flight directions in the flock. Comparing flocks of different densities, the range of interactions that contribute to the energy involves a fixed number of (topological) neighbors, rather than a fixed (metric) spatial range. Comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long ranged correlations among the fluctuations in flight direction.

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Novel type of phase transition in a system of self-driven particles

Andras Czirok, Ofer Sochet, Tamás Vicsek … (2006)

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation (\(\eta\)) added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, \(| {\bf v}_a | =0\)) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since \(| {\bf v}_a |\) is found to scale as \((\eta_c-\eta)^\beta\) with \(\beta\simeq 0.45\).

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General Theory of Spin-Wave Interactions

Freeman J. Dyson (1956)

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Weak pairwise correlations imply strongly correlated network states in a neural population

Gasper Tkacik (2006)

Biological networks have so many possible states that exhaustive sampling is impossible. Successful analysis thus depends on simplifying hypotheses, but experiments on many systems hint that complicated, higher order interactions among large groups of elements play an important role. In the vertebrate retina, we show that weak correlations between pairs of neurons coexist with strongly collective behavior in the responses of ten or more neurons. Surprisingly, we find that this collective behavior is described quantitatively by models that capture the observed pairwise correlations but assume no higher order interactions. These maximum entropy models are equivalent to Ising models, and predict that larger networks are completely dominated by correlation effects. This suggests that the neural code has associative or error-correcting properties, and we provide preliminary evidence for such behavior. As a first test for the generality of these ideas, we show that similar results are obtained from networks of cultured cortical neurons.