During the 1950s Hodgkin and Huxley came up with a set of differential equations which mimic the physics of neural activity - based on experiments with the giant squid axon.
In essence, a neuron doesn’t fire until a threshold input stimulus is reached and, after firing, neurons exhibit a relaxation time, which prevents them from firing again for a short time.
A simplified version of the (four dimensional) Hodgkin-Huxley model is the (two dimensional) Fitzhugh -Nagumo [1] oscillator:
In the above, x is the neuron membrane voltage, a, b, c and k are constants and y is the 'recovery variable' for the neuron membrane potential. The membrane voltage (x) has a cubic non-linearity.
Fitzhugh-Nagumo oscillators can be coupled together - Sprott considers the following [2] coupling of two Fitzhugh-Nagumo oscillators - the stimulus for the first is some constant (k) times the membrane voltage of the second (x2); the stimulus for the second is some constant (k) times the membrane voltage of the first (x1):
Note: there appears to be a typographical error in Sprott's equation (6.10) - the third equation should (I think) be as I've written above - i.e. y2 rather than y1 appearing on its right hand side.
Here's my patching for the coupled equations:
The following plots were obtained using the parameter values / initial conditions given by Sprott [2] (i.e. a = 0.4, b = 0.5, c = 0.01 and k = -1). The plots show the first ten minutes of the trajectories - using the 0.2 µF integration capacitors throughout. The red plot shows y2 vs. x2; the black plot shows y1 vs. x1.
With the computer in FAST mode I captured the following (y1 vs. x1)...just before my Tektronix 465 blew up(!).
It's interesting to watch the trajectory unfold with the computer running in slow time (0.2 µF integration capacitors = 0.2 second time constant), on the plotter. The pen whizzes around the perimeter from the initial condition position, settling down at (almost) a point (towards bottom left - resting state - bright splodge on oscilloscope display); the trajectory then slowly spirals outwards (initially very slowly) from this resting point, eventually gaining speed (and eventually reaching the firing threshold / stimulus) and going around the perimeter again (quickly), and so on.
Sometimes the fast part of the trajectory follows the outer path / orbit, followed immediately by a second slightly inner fast path / orbit, before returning to the 'resting spiral' (equilibrium resting potential). The fast moving outer parts of the trajectory corresponds to the neuron firing; the spiral part corresponds to the equilibrium / resting part. I guess the second fast orbit which is, on occasion, observed corresponds to a relative refractory (see below) part of the trajectory.
Reference [3] gives a good depiction of the trajectory and its elements, reproduced below. The neuron fires (active part of trajectory); immediately followed by a part in which it cannot be stimulated, no matter how great a stimulus is applied (absolute refractory), followed by a 'relative refractory' part (in which a second firing is possible - but only given enough stimulus) - followed by the resting phase.
References
[1] FitzHugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1:445-466.
[2] Sprott J.C. Elegant Chaos: Algebraically Simple Chaotic Flows, pp. 123 - 125.
[3] http://www.scholarpedia.org/article/FitzHugh-Nagumo_model
In essence, a neuron doesn’t fire until a threshold input stimulus is reached and, after firing, neurons exhibit a relaxation time, which prevents them from firing again for a short time.
A simplified version of the (four dimensional) Hodgkin-Huxley model is the (two dimensional) Fitzhugh -Nagumo [1] oscillator:
Fitzhugh-Nagumo oscillators can be coupled together - Sprott considers the following [2] coupling of two Fitzhugh-Nagumo oscillators - the stimulus for the first is some constant (k) times the membrane voltage of the second (x2); the stimulus for the second is some constant (k) times the membrane voltage of the first (x1):
Note: there appears to be a typographical error in Sprott's equation (6.10) - the third equation should (I think) be as I've written above - i.e. y2 rather than y1 appearing on its right hand side.
Here's my patching for the coupled equations:
 |
| Richard FitzHugh with analog computer, ca. 1960, from [3]. |
 |
| Analog computer, ca. 2015. Patched for coupled FitzHugh - Nagumo system. |
The following plots were obtained using the parameter values / initial conditions given by Sprott [2] (i.e. a = 0.4, b = 0.5, c = 0.01 and k = -1). The plots show the first ten minutes of the trajectories - using the 0.2 µF integration capacitors throughout. The red plot shows y2 vs. x2; the black plot shows y1 vs. x1.
 |
| 10 minutes' worth of trajectory. Green dots mark start / initial condition point in each case (red pen took time to get going!). |
With the computer in FAST mode I captured the following (y1 vs. x1)...just before my Tektronix 465 blew up(!).
 |
| Attractor for the coupled FitzHugh - Nagumo system, as per [2], figure 6.7. (1 V/div both axes.) |
It's interesting to watch the trajectory unfold with the computer running in slow time (0.2 µF integration capacitors = 0.2 second time constant), on the plotter. The pen whizzes around the perimeter from the initial condition position, settling down at (almost) a point (towards bottom left - resting state - bright splodge on oscilloscope display); the trajectory then slowly spirals outwards (initially very slowly) from this resting point, eventually gaining speed (and eventually reaching the firing threshold / stimulus) and going around the perimeter again (quickly), and so on.
Sometimes the fast part of the trajectory follows the outer path / orbit, followed immediately by a second slightly inner fast path / orbit, before returning to the 'resting spiral' (equilibrium resting potential). The fast moving outer parts of the trajectory corresponds to the neuron firing; the spiral part corresponds to the equilibrium / resting part. I guess the second fast orbit which is, on occasion, observed corresponds to a relative refractory (see below) part of the trajectory.
Reference [3] gives a good depiction of the trajectory and its elements, reproduced below. The neuron fires (active part of trajectory); immediately followed by a part in which it cannot be stimulated, no matter how great a stimulus is applied (absolute refractory), followed by a 'relative refractory' part (in which a second firing is possible - but only given enough stimulus) - followed by the resting phase.
 |
| Phase portrait and physiological state diagram of FitzHugh-Nagumo model, taken from [3]. |
References
[1] FitzHugh R. (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophysical J. 1:445-466.
[2] Sprott J.C. Elegant Chaos: Algebraically Simple Chaotic Flows, pp. 123 - 125.
[3] http://www.scholarpedia.org/article/FitzHugh-Nagumo_model
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