Statistical Mechanics and Equilibrium Properties – Small Neural Ensembles

Statistical Mechanics of Small Neural Ensembles – Commentary on Tkačik et al.

In a series of related articles, Gašper Tkačik et al. (see references below) investigated small (10-120) groups of neurons in the salamander retina, with the purpose of estimating entropy and other statistical mechanics properties.

They provide the following interesting results:

1. Simple scheme for entropy estimation in undersampled region (1), given that only a small fraction of possible states can be experimentally obtained during any trials,
2. Improved distribution model (2) over basic nearest-neighbor interacting Ising model, by identifying intrinsic inhomogeneity in activity distribution, so that a combination of two distinct activation models best matches with experimental results,
3. Emergence of well-defined, smooth relationship between entropy and entropy (3), together with estimating thermodynamic parameters as functions of ensemble size (from 20 – 160 neurons), and
4. Emergence of criticality (divergence in heat capacity) (4) with growth in ensemble size (from N = 20 – 120 neurons), together with nonlinear growth in heat capacity as ensemble size increases.

Update: Friday, Nov. 4, 2016:

A recent paper by Nonnenmacher et al. (see citation below) refutes the basic premise of criticality in these small neuronal groups. This is worth a read, and a reread. I’m putting the reference here so that I can find it later.

1. Marcel Nonnenmacher, Christian Behrens, Philipp Berens, Matthias Bethge, and Jakob H. Macke, Signatures of criticality arise in simple neural population models with correlations. arXiv:1603.00097v1 [q-bio.NC] (29 Feb 2016). pdf

1. MJ Berry 2nd, G Tkačik, J Dubuis, O Marre, and R Azeredo da Silveira, A simple method for estimating the entropy of neural activity. J Stat Mech, 04: P03015 (2013). (doi:10.1088/1742-5468/2013/03/P03015). [site] [pdf]
2. G Tkačik, O Marre, D Amodei, E Schneidman, W Bialek, MJ Berry 2nd, Searching for collective behavior in a large network of sensory neurons. PLoS Comput Biol 10 (1): e1003408 (2014). doi:10.1371/journal.pcbi.1003408. [site/abstract] [pdf]
3. G Tkačik, O Marre, T Mora, D Amodei, MJ Berry 2nd, and W Bialek, The simplest maximum entropy model for collective behavior in a neural network. J Stat Mech P03011 (2013). (doi:10.1088/1742-5468/2013/03/P03011) [site] [pdf] [arXiv]
4. G Tkačik, T Mora, O Marre, D Amodei, MJ Berry 2nd, and W Bialek, Thermodynamics for a network of neurons: signatures of criticality. arXiv.org (2014): 1407.5946. [arXiv]