Thermodynamics and signatures of criticality in a network of neurons.
The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.
Institute of Science and Technology Austria, A-3400 Klosterneuburg, Austria;
Gašper
G
Tkačik
Laboratoire de Physique Statistique, CNRS, Université Pierre et Marie Curie (UPMC) and l'École Normale Supérieure, 75231 Paris Cedex 05, France;
Thierry
T
Mora
Institut de la Vision, UMRS 968 UPMC, INSERM, CNRS U7210, F-75012 Paris, France;
Olivier
O
Marre
Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544; Princeton Neuroscience Institute, Princeton University, Princeton, NJ 08544;
Dario
D
Amodei
Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544; Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544;
Stephanie E
http://orcid.org/0000-0001-6211-6293
SE
Palmer
Princeton Neuroscience Institute, Princeton University, Princeton, NJ 08544; Department of Molecular Biology, Princeton University, Princeton, NJ 08544;
Michael J
MJ
Berry
Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544; Initiative for the Theoretical Sciences, The Graduate Center, City University of New York, New York, NY 10016 wbialek@princeton.edu.
William
W
Bialek
D000465:Algorithms; D000818:Animals; D001921:Brain / Q000502:physiology*; D019277:Entropy; D006358:Hot Temperature; D008959:Models, Neurological; D015233:Models, Statistical; D009010:Monte Carlo Method; D009415:Nerve Net; D009474:Neurons / Q000502:physiology*; D011336:Probability; D015203:Reproducibility of Results; D012160:Retina / Q000502:physiology; D013816:Thermodynamics; D014562:Caudata
Monte Carlo; correlation; entropy; information; neural networks