4.4 The Physics of Neural Circuits

The physics underlying excitable phenomena of the nervous system and its connected muscular system (biophysics) is based on electrochemistry. In addition, mathematical models provide a powerful way to investigate how neural circuits operate in physiological conditions. These models are based on the biophysical properties of neurons and constitute a growing field of computational neuroscience.

The mathematical features of the propagation of activity within a nerve cell and within a neural circuit are well captured by equations that follow the reaction-diffusion properties, described for excitable media (BZ equations, described above: Section 1.6). When an action potential is generated by a transient depolarisation of the membrane (the reaction), it influences neighbouring patches of the cell membrane by a passive propagation of the depolarising current (diffusion). In turn, this triggers a new action potential in the neighbouring patch with consequent propagation of the action potential along the full length of the axon. This is how neurons carry electrical signals for significant distances. 

The Hodgkin and Huxley (H&H) equations, which explain well the electrical nature of neural activity (see Section 4.2), can be modified to transform the electrical models of quiescent neurons into models of pacemaker neurons with spontaneous oscillations¹. Richard Fitzhugh and Jinichi Nagumo recognised the similarities between Van der Pol electrical oscillators and oscillating neurons. In 1961, they showed that the Hodgkin-Huxley equations could be simplified to equation that contains the Van der Pol equation², now called the Fitzhugh-Nagumo or Bonhoeffer-Van der Pol equation. Furthermore, the periodic activity of individual neurons treated as oscillators can be portrayed as space-phase plots like those developed for orbits of planets and pendula. These portrayals reveal oscillating neuronal systems with periods of regular or chaotic activity³.

Simplified models of neurons called integrate and fire neurons devised by FitzHugh and Nagumo allowed relatively easy exploration of the patterns of activity of interconnected neurons in mathematical models of neural circuits⁴. These mathematical models are commonly used to describe the spatio-temporal dynamics of interconnected neurons firing action potentials, particularly under conditions in which they oscillate and synchronise their activity to generate a rich variety of firing patterns. 

There is a reasonable expectation that there is some relation between the wiring of a neural circuit and its functional states⁵. However, the spatio-temporal patterns of activity vary significantly, depending on the strength of the synaptic connections (ie, the strength of their coupling). When the strength of the synaptic transmission is greater, the circuit behaves as a strongly coupled, ‘hardwired’ system and the patterns of activity are more stable and stereotyped. 

When the nerve cells of a neural circuit are weakly coupled (ie, ‘not hard wired’), they can generate spatio-temporal patterns of activity that resemble chaotic behaviour, with traveling, concentric and spiral waves like those described in the BZ chemical reactions. Even from initially random firing of weakly interconnected neurons, coherent and chaotic firing emerges, according to the rules of active media with spreading of activity as circumferential, travelling, and spiral waves with annihilation of colliding waves.

Simple neural circuits in a worm

How the ‘wiring’ (architecture) of a neural circuit relates to its behaviour (function) has been investigated in detail in the more simple nervous systems of invertebrates. A classic example is Caenorhabditis elegans, a small nematode worm which consists of a small number of cells (about 1000) of which only about 300 are neurons. Its neural wiring has been studied in detail by Sydney Brenner, who won the Nobel prize for his studies in 2002. The neural circuits of this worm underly complex behaviours, including navigation, responses to ‘social’ chemicals (pheromones) and can change through learning.

In addition to classic synaptic transmission, the more complex process of multiple neurotransmission involving neuropeptides occurs also in Caenorhabditis.  Recent investigations demonstrate a subtle modulatory role of these transmitters which may act at some distance from the synapses suggesting a “wireless network whose organization differs in important ways from hard wired circuits” that could help explain the variable behaviour of individual worms⁷. Indeed, measuring signal propagation in these circuits better predicts the neural dynamics underlying spontaneous activity in the worm than do models based solely on ’hard wiring’ derived from the neuroanatomy. These experiments further indicate how densely connected networks of neurons can communicate over long distances (extrasynaptic transmission), rather than just across narrow synapses⁸.

These studies confirm the idea that even simpler organisms with identical neural wiring across individuals nevertheless can generate different complex behaviours. Conversely similar behaviours can be generated by very different neural wiring patterns.

A caution…

Caution needs to be raised about computational modelling. We have already established that deterministic rules in non-linear dynamical systems do not imply predictability. In real neuronal circuits, the absolute details of the initial values and boundary conditions usually cannot be established. 

Furthermore, models used to simulate states of neural circuits suitable to understand the emergence of specific functions become inadequate at some higher degree of detail. Therefore, fully predictive mathematical models of larger-scale biomechanical and neurophysiological events cannot be achieved. It follows that the sequence of states of the underlying neural circuits cannot be fully predicted. These issues of the relationship between the architecture and function of neural circuits and the impossibility to establish their initial conditions will become relevant when I will discuss the problems of decision making and ‘free will’ later in this essay.

Before discussing the patterns of activity of neural circuits of the brain, I will describe next the fundamental architecture of neural circuits of the vertebrate nervous system. 

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1. These equations were originally developed to explain pacemaker potentials in heart muscle cells, see DA Noble (1962): A modification of the Hodgkin-Huxley equations applicable to Purkinje fibre action and pacemaker potentials. Journal of Physiology 160, 317-352. ↩︎
2. Richard FitzHugh (1961): Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal 1, 445-466. ↩︎
3. For example, see: X-J Wang (1994): Multiple dynamical modes of thalamic relay neurons: Rhythmic bursting and intermittent phase-locking. Neuroscience 59, 21-31.
   T Elbert et al (1994): Chaos and physiology: deterministic chaos in excitable cell assemblies. Physiology Reviews 74, 1-47.
   Michael Breakspear (2017): Dynamic models of large-scale brain activity. Nature Neuroscience 20, 340-352. ↩︎
4. For more see: W Gerstner et al (2014): Neuronal Dynamics – From Single Neurons to Networks and Models of Cognition. Cambridge University Press, especially https://neuronaldynamics.epfl.ch/online/Ch1.S3.html;
   Also: http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model
   For examples, see Bard Ermentrout (2001): Neural Networks as Spatio-Temporal Pattern-Forming Systems. Reports on Progress in Physics 61(4) 353-430. ↩︎
5. For example, see: https://nba.uth.tmc.edu/neuroscience/m/s1/introduction.html ↩︎
6. For more, see R Henkel et al (2002): Synchronizing assemblies perform magnitude-invariant pattern detection. Neurocomputing 44-46, 429-433. ↩︎
7. L Ripoli-Snachèz et al (2023): The neuropeptidergic connectome of C. elegans. Neuron 111, 3570-3589. ↩︎
8. F Randi et al (2023): Neural signal propagation atlas of Caenorhabditis elegans. Nature 623, 406-420.
   See also data from neural circuits in crabs: EM Cronin et al (2024): Modulation by Neuropeptides with Overlapping Targets Results in Functional Overlap in Oscillatory Circuit Activation. Journal of Neuroscience 44, e1201232023. ↩︎

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