Voltage-controlled Oscillations in Neurons

• Dr. Frank Hoppensteadt, Courant Institute of Mathematical Sciences, NYU, New York, NY

VCON is an acronym for the voltage controlled oscillator neuron model. It has the form \[\tau\ddot\theta + F(\dot\theta) + A \sin\theta = \omega \] where \(\dot\theta\) corresponds to the membrane potential in an action potential generating region of an axon.

Introduction

Hodgkin and Huxley discovered that voltages control ionic currents in nerve membranes. This led them to describe electrical activity in a neuronal membrane patch in terms of an electronic circuit whose characteristics were determined using empirical data (Hodgkin et al. 1952). Due to the complexity of this system, a number of useful heuristics for the Hodgkin-Huxley circuit have been devised, which are reviewed in (Hoppensteadt, 2012). The heuristic presented here is based on a phase-amplitude analysis of the van der Pol model of neural activity, which revealed a rich structure of phase-locking behavior (Flaherty and Hoppensteadt 1978). This work motivated experiments with forced rhythms in squid axons that revealed remarkably similar phase-locking behavior (Guttman et al. 1980). The VCON heuristic emerged from this (serial) collaboration in 1980; it was also found to represent phase-locked loop circuits in electronics (Horowitz and Hill 1989). VCONs share various behaviors with neurons, but direct frequency analysis is possible, which is not the case for other models theretofore (Hoppensteadt 1997). While a VCON is consistent with numerous observations in neuroscience, it is also constructible as an electronic circuit on scales ranging from nano-scale spin torque oscillators (Macia et al.) to phase-locked loop integrated circuits (Hoppensteadt et al. 1997). This article pertains to the mathematical dynamics of a VCON rather than to its uses in neuroscience or engineering.

Figure 1: Top: The Integrate-and-fire circuit was introduced and studied by Schmitt (Schmitt et al (1931)). SW denotes a voltage controlled switch that opens and closes when V hits 0 and a threshold, respectively. Middle: The FitzHugh-Nagumo circuit incorporates a tunnel diode (TD), which is a negative differential resistance element, as an escapement. It recreates and extends important aspects of van der Pol's model. The channel marked L has a resistor, an inductor and a battery in series. Bottom: The VCON as discussed in the text incorporates a negative differential resistance element (NDR) as an escapement and a voltage controlled oscillator H as a homeostatic mechanism.

Principles

The VCON is based on four principles of neuroscience:

• Neural membranes can separate charges;
• a resting membrane potential is sustained by cell metabolism;
• homeostatic mechanisms stabilize the resting potential; and,
• escapements destabilize the resting potential, sometimes leading to action potentials.

These principles have inspired the design of many electronic circuits, among them are the Hodgkin-Huxley model (Hodgkin and Huxley 1952) and those shown in Figure 1. However, only the VCON directly models relationships between voltage and frequency. The VCON is used as a model of an action potential generator region of a neuron's membrane in synthesizing and analyzing networks.

The model components are:

• for the separation of charge, a capacitor,
• for the resting potential, a current source (a battery and resistor),
• for the homeostatic mechanism (comparable to the role played by potassium channels in the Hodgkin-Huxley model), a voltage controlled oscillator (Hoppensteadt 1997),
• for the escapement (comparable to the role played by sodium channels in the Hodgkin-Huxley model), a negative differential resistance (NDR) device (e.g., see FitzHugh-Nagumo model and Schmitt et al. (1931)).

The circuit depicted at the bottom of Figure 1 shows the arrangement of these elements. The current \(I\) is divided among the separate channels according to their own \(IV\)-characteristics:

• The current through the capacitor is \(C \dot V\ ;\)
• the current through the escapement is \(I_N=f(V)\) where \(f\) is an N-shaped function (it describes an NDR device); and,
• the current through the homeostatic mechanism \(H\) is \(I_H=\alpha\sin\left(\gamma\int_0^t V(t')\,dt'\right)\) (Horowitz and Hill 1989).

Since \(\int V\) appears only inside a periodic function, we define it to be a new angle variable \(\theta(t)=\gamma\int_0^t V(t')\,dt'\ .\) As a result, we have that the instantaneous frequency of current through the homeostatic junction is proportional to \(V\ :\) i.e., \(\dot\theta(t) = \gamma V(t)\) (here and below \(\dot\theta = d\theta/dt\ ,\) etc.). Similar formulas relating frequency and voltage arise in electronics on scales ranging from quantum mechanical Josephson junctions, to phase-locked loop integrated circuits, to rotating electrical machinery of all sizes, and to turbines on regional power grids.

The model of the VCON circuit is based on Kirchhoff's law that balances currents, \(I=C\dot V + I_H+I_N\ ,\) and Ohm's Law that describes the current source, \(R I=E-V\ .\) Using the fact that \(\gamma V=\dot\theta\ ,\) we have \[\dot\theta = \gamma V\] \[\tau\dot V = E-V-R(f(V)+\alpha\sin\theta)\] where \(\tau=R C\) is a time constant. Substituting the first equation in the second gives a second order differential equation for \(\theta\ :\) \[\tau\ddot\theta+F(\dot\theta)+A\sin\theta=\omega\] where \(A=R\alpha\gamma \ ,\) \(F(\dot\theta)=\dot\theta+R\gamma f(\dot\theta/\gamma)\) and \(\omega=\gamma E\) (these terms have units of radians/sec). This equation has familiar components: Linearizing \(\sin\theta\approx\theta\) gives Rayleigh's equation (which is essentially equivalent to van der Pol's equation); and, taking \(f\equiv 0\) gives a pendulum equation. While these equations are well known separately, the combined model has certain interesting and useful new features, such as supporting both saddle-node on invariant circle (SNIC) and Andronov-Hopf bifurcations and exhibiting coexistent stable oscillations.

Figure 2: Simulation of \(\ddot\theta+0.5\,\dot\theta\,(\dot\theta^2-0.5)+\sin\theta=0.6\ .\) Phase portrait of two stable oscillations demonstrating coexistence of stable bounded and unbounded oscillations on the cylinder. Note that the horizontal axis has been shifted by 1.0 to facilitate plotting. These images are explained in the text.

Sub- and Super-threshold oscillations

The coexistence of stable super- and sub-threshold oscillations is demonstrated in Figure 2. The top oscillation, referred to as being a running periodic solution, wraps around the cylinder \(\mathcal{C} = \{(\theta\, \textrm{ mod }\, 2\pi,V)\}\ ,\) and the lower one remains on one sheet of the cylinder. There is a saddle point lying to the right of the lower oscillation at \((\pi/2+\arcsin 0.6,0.0)\ ,\) as indicated by an asterisk. The separatrix entering it from above is bounded above by the upper oscillation and must come from \(V = -\infty\ .\) The lower oscillation resulted from an Andronov-Hopf bifurcation. In addition, as \(\omega\) increases through the value \(A\ ,\) a saddle-node bifurcation occurs, after which the lower oscillation disappears and either one or two running periodic solutions exist (See Hoppensteadt 2006).

There is an unstable equilibrium within the lower orbit at \((\arcsin 0.6, 0.0)\ ,\) as indicated by an asterisk. The output currents in these two cases are proportional to \(\sin\theta(t)\ ;\) in the former case (running periodic solution), the output is a fully developed sinusoid, in the latter (bounded oscillation) the current oscillates but is bounded away from \(\pm 1\ .\)

Figure 3: Simulation of \(\rho\) for \(\ddot\theta+\dot\theta+3(1+\cos 2\pi t) \sin\theta=2\pi \omega\) The output frequency \(\rho\approx \theta(200)/400\pi\) is plotted for each of 200 equally spaced values of \(\omega\) between 0.5 and 0.85. The units in this graph are hz vs. hz. Phase locking is apparent for \(\rho\) = 1/1, 1/2, 2/3, 3/4, 3/5, etc., as suggested by Farey's sequence (Flaherty and Hoppensteadt 1978). This demonstrates that VCON has many natural modes with which external signals can resonate.

External Forcing

External signals can be brought into VCON in several ways. Studies of phase-locked loops suggest parametric forcing (e.g., \(A \propto \cos\mu t\)) and additive inputs (e.g., \(\omega = \Omega(t)\)), which are described first. Parametric forcing of the escapement is demonstrated second. These modified circuits are constructible using off-the-shelf electronics, and they have analogs in neuroscience (Hoppensteadt 1997).

The output frequency (hz) is \[\rho=\lim_{t\to\infty} \frac{\theta(t)}{2 \pi t}=\lim_{t\to\infty}\frac \gamma{2\pi t} \int_0^t V(t')\,dt'.\] As a result, the output current is proportional to \(\sin(\rho t + \phi(t))\) where \(\phi(t)\) is a (bounded) oscillatory phase deviation (Hoppensteadt 1997). The frequency-response pattern of the VCON \[\ddot\theta+\dot\theta+3(1+\cos 2\pi t) \sin\theta=2\pi \omega\] illustrates phase-locking. Figure 3 suggests that \(\rho\) will be a monotone, continuous function of \(\omega\ ,\) and that it will form a staircase whose treads correspond to phase-locked responses. While this behavior is stable, it is shown here for one set of initial data ((0,0), in this case), other initial data can result in other phase-locking patterns (Flaherty and Hoppensteadt 1978).

Figure 4: Forcing the escapement. At the bottom (on the line labeled 0), is a plot of \(p_1(t)\ ;\) this comprises two pulses separated by 3.5 (leading edge to leading edge). Above that is plotted the output (the current through H) \(I_1(t)\) - no action potentials occur in this case. The plot above that line is the phase of this current, \(\theta_1(t)\ .\) The line labeled 20 plots \(p_2(t)\) where the pulses are separated by 4. In this case, \(\theta_2(t)\) is pushed over the top and increases by \(2\pi\ .\) One action potential results (its shape should be compared to that in http://en.wikipedia.org/wiki/Action_potential). On the line labeled 40, the two pulses in \(p_3(t)\) are separated by 8. As in the first case, no action potential results. The horizontal scale is roughly equivalent to milliseconds and the vertical scale is roughly equivalent to millvolts.

The frequency \(\rho\) suggests that there is an ongoing sustained oscillation in the electronic circuit. However, since the VCON is pendulum-like, only a few pushes are needed to force it over the top to execute a full oscillation. But, the timing of these pushes, which for the circuit are electrical spikes, must be precise. The critical inter-spike interval (ISI) is indicated by the natural period \(2\pi/\rho\ .\) Just a few correctly timed spikes will cause a VCON to produce a full oscillation, similar to the resonate and fire phenomenon. This rapid acquisition time is an important aspect of VCON, as it shares this property with neurons that are known to respond to a few precisely timed input spikes (Izhikevich et al. 2003). The final simulation demonstrates this sensitivity to timing of inputs.

The simulation in Figure 4 demonstrates the response when properly spaced inputs are applied to the escapement of the VCON \[\ddot\theta+\left(1+p(t)(\dot\theta^2-10)\right)\dot\theta+3\sin\theta=1.2\pi,\] where \(p(t)\) denotes a train of two pulses. Three cases are simulated here: (1) Short inter-spike interval (ISI = 3.5), (2) intermediate (ISI = 4.0), and (3) long (ISI = 8.0). The oscillator responds with a full spike-generating oscillation in the second case, but not in the first or third. This demonstrates the phenomenon of sensitivity only to correctly timed input spikes.

References

• Hodgkin A.L., Huxley A.F. (1952) A quantitative description of membrane current and its applications to conduction and excitation in nerve. J. Physiol. 117:500-544.
• Hoppensteadt F. (2012) Heuristics for the Hodgkin-Huxley system, Math. Biosci. http://dx.doi.org/10.1016/j.mbs.2012.11.006
• Flaherty J.E., Hoppensteadt F.C. (1978) Frequency entrainment of a forced van der Pol oscillator, Studs. Appl. Math. 58:5-15
• Guttman R., Feldman L., Jacobsson E. (1980) Frequency entrainment of squid axon membrane, J. Membrane Biol. 56:9-18
• Hoppensteadt F.C. (1997) Introduction to Mathematics of Neurons, Modeling in the frequency domain (2nd ed.) Camb. U. Press
• Macia F., Kent A., Hoppensteadt F. (2011) Spin-wave interference patterns created by spin-torque nano-oscillators for memory and computation, arXiv, no. 1009.4116 (tu8mk). Nanotechnology 22:095301. DOI:10.1088/0957-4484/22/9/095301.
• Hoppensteadt F.C., Izhikevich E.M. (1997) Weakly Connected Neural Networks, Springer-Verlag, New York.
• Hoppensteadt F.C. (2006) Biologically inspired circuits, Int. J. Bifurcation and Chaos, October, 2006.
• Horowitz P., Hill W. (1989) The Art of Electronics, 2nd ed., Camb. U. Press.
• Izhikevich E.M., Desai N.S., Walcott E.C., Hoppensteadt F.C. (2003) Bursts as a unit of neural information: Taking advantage of resonance, Trends in Neuroscience, 26:161-167.
• Schmitt O., Schmitt, F. (1931) The Nature of the Nerve Impulse, Am. J. Physiol., Vol. 97 # 2.

Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

External links

• Frank Hoppensteadt's website

See also

Dynamical Systems, Excitability, Periodic Orbit, Phase Space, Phase-Locked Loop, Resonance, Synchronization, Voltage Controlled Oscillator

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Accepted on: 2006-11-06 20:13:52 GMT