The Ising Model: From Neuron Firing to Sea Ice Melting

Magnetism, Neuron Firing, Arctic Sea-Ice Melting. On the surface, it would seem pretty ludicrous to suggest that all these phenomena can be described by a theoretical framework developed in the 1920’s. But it is true! Yes, all of these systems can be studied and described using the Ising model: a model first proposed by Wilhelm Lenz in 1920 but was solved in the one-dimensional case by Ernst Ising in his 1925 thesis [8].

A lattice arrangement of objects undergoing nearest-neighbour interactions can describe very accurately a large array of physical, chemical AND biological systems. The most simplistic, yet poplar, theoretical framework for this arrangement is the Ising Model. Despite Ising originally rejecting the very simple binary based framework, the model led on to explain all types of phenomena of a system (to a good approximation) including gas-liquid phenomena, magnetic Curie points, order-disorder transitions in alloys and phase separation in liquid mixtures. [1]

But what exactly is the ideas behind the Ising model?

The model has spin variables σ_i , where i represents the i^th   lattice point. These spins take on two discrete values: σ_i = ±1 . In the case of ferromagnetism, each particle has either a spin-up σ_i = +1 or a spin-down σ_i = -1 on each site in conjunction with the externally applied magnetic field. Even though the theory was originally formulated for ferromagnetism, it can be extended to other systems. For examples if we wanted to describe a liquid composed of a mixture of two different kinds of molecules, one of the molecules could be labelled as σ_i = +1 and the other with σ_i = -1. As this is a binary system there are 2^N configurations of the system for N lattice sites. [1][6]

Figure 1: An example of a configuration of a system in which N = 25 [6]

The assumption of the Ising model is that particles at each site can only exert short-range forces on each other. This interaction energy tends to align the spins of neighbouring entities. Under this assumption, this interaction could lead to spontaneous magnetization (whereby all/most of the spins are aligned in the material even without the presence of an external magnetic field) or phase separation in the case of the mixed liquid mentioned above! [6]

The Hamiltonian of the d-dimensional Ising model is given by

where the sum <ij> is over the nearest neighbours. H_j represents the external magnetic field and J_ij is the exchange constant between the  i^th   and the j^th particles. The magnitude of the exchange constant specifies the strength of the spin interactions. From the Hamiltonian, it can be observed that under the assumption of the short-range interaction of the framework, the more neighbouring atoms having the same spin, the lower the energy of the overall system.[2] [6]

The mathematical problem is to solve analytically the partition function of the system! With the partition function, you can then calculate all the thermodynamic functions describing the system. [1] For example, the entropy, internal energy, Gibbs Free energy, Helmholtz free energy etc.

But what has this got to do with neuron firing and arctic sea-ice melting?

In his paper in 1982[3], John Hopfield made the connection that memory modelling (i.e., studying the activity of neurons in the brain) can be described (up to first approximation) using the Ising model. He noted that neurons can be characterised into two states: firing and not-firing, which he could denote with the two spin states required in the Ising model (they were +1 and 0 respectively for each neuron state). Neurons also influenced one another via the auto-associative memory process (which enables human brains to recall a complete concept from an incomplete set of clues)[5], and hence the strength of a connection between two neurons could be quantified by the coupling-constant J_ij = +1. Memory processes could also be perceived as a processes governed by the principle of maximum entropy. In summary, neuron interactions make the brain formulate the nearest memory once prompted by a clue.

The groundbreaking discovery of the “Hopfield-Network” has led to numerous applications: from seizure treatment all the way to studying memory loss in patients with dementia.[5]

As well as having applications to biological systems and processes, natural systems can be studied using the Ising model. One such system studied is the “melt ponds” in the cold Arctic landscape. Melt ponds are basically a pond of liquid water (deriving mostly from melted snow) on the surface of ice, and they are very common in the Arctic![5]

Figure 2: The above figure shows the terrain of a part of the the Chukchi Sea in the Arctic. One can see the melt ponds among the ice! [7]

It is common knowledge that natural systems tend towards their lower energy states. This was known to mathematician Kenneth M Golden, who made the connection between the Ising model and studying the how geometrical patterns of melt ponds formulated in the landscape of the Arctic [2]. In the study led by Golden [2], the team discovered that they could characterize each site in a virtual landscape by two states: water (σ_i = +1) and ice (σ_i = -1) and these “islands” of the same spins looked like melt ponds! Each site was also assigned a specific height to mimic the geometry of the Arctic landscape.

In the case of ferromagnetism, we know the spins of magnetized atoms tend to align with those of their neighbours. This can be related to the melt ponds: a pond surrounded by ice is more likely to freeze over and a patch of ice surrounded by water is more likely to melt. This is just another version of the nearest-neighbour interaction assumption of the Ising model! However in the case that the nearest neighbours did not have a majority spin, a “tie-breaker” rule was implemented into the algorithm. This rule describes the tendency of the water to fill depressions in the landscape, i.e., to go to as low an energy state as possible! [2] [5]

The model of these melt ponds was incredibly accurate and very elegant due to its underlying simplicity, and it could possibly make real world projections for global climate models. [5]

This elegant yet simplistic theoretical framework unites a diverse and colourful array of applications, and really shows us through mathematics how connected the natural world is. Not bad for a theory which was deemed a “not sufficient to explain ferromagnetism” by Heisenberg in 1928![1]

References

[1]S. G. Brush, History of the Lenz-Ising Model, Rev. Mod. Phys 39 (1962) 883-893

[2]Ma Yi-Ping,
Sudakov Ivan,Strong Courtenay, Kenneth M. Golden, Ising model for melt ponds on Arctic sea ice NEW JOURNAL OF PHYSICS. 1367-2630. JUNE 2019. 21.6.063029, 10.1088/1367-2630/ab26db, WOS:000503044800001

[3] Hopfield JJ. Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci U S A. 1982 Apr;79(8):2554-8. doi: 10.1073/pnas.79.8.2554. PMID: 6953413; PMCID: PMC346238.

[4] American Meteorological Society (AMS) 2022 “Melt Pond”, 10/04/2023, https://glossary.ametsoc.org/wiki/Melt_pond

[5] Leila Sloman, Magnet and Neuron Model Also Predicts Arctic Sea Ice Melt, SCIENTIFIC AMERICAN, a Division of Springer Nature America, Inc. Date: Jul 24, 2019

[6] Pathria, R.K. and Beale, P.D. (2011) Statistical Mechanics. 3rd Edition, Elsevier, Singapore, 583-586. http://dx.doi.org/10.1016/B978-0-12-382188-1.00015-3

[7] Image Credit: NASA/Kathryn Hansen, ICESCAPE Arctic Sea Ice Mission, Last Updated: Aug 7, 2017, Editor: NASA Content Administrator, https://www.nasa.gov/multimedia/imagegallery/image_feature_1921.html

[8] E. ISING, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift für Physik 31 (1925) 253-258.