Models for Spiking Neurons: Integrate-and-Fire Units and Relaxation Oscillators

Abstract

Relaxation oscillators are nonlinear electronic circuits that produce a repetitive non-sinusoidal waveform when sufficient voltage is applied. In this fashion, they are reminiscent of integrate-and-fire neuron models, except that they also include components with hysteresis, and thus require no threshold rule to determine when an impulse has occurred or to return the voltage to its reset value. Here, I discuss the pros and cons of teaching elementary neurophysiology using first-order linear integrate-and-fire neurons versus relaxation oscillator circuits. I suggest that the shortcomings of both types of models are useful in order to foster a critical understanding of the neurophysiology underlying the firing dynamics of biological neurons.

Keywords: integrate-and-fire model; python programming; relaxation oscillator; teaching lab; transistor.

Figure A1

Photograph of a prototyping board with the circuit shown in Figure 8. Note that pin 2 of the 2n2222 is open (not connected to anything); it is actually bent upward toward the ceiling. The two yellow wires at the bottom of the image are connected to the potentiometer. The red and black leads at the top of the image are connected to the oscilloscope. The red and black wires at the left of the image are from the batteries.

Figure A2

Photograph of the experimental apparatus used to visualize the spiking activity of the resonance oscillator depicted in Figures 8 and A1. Note that 2 9V batteries are connected in series to provide power. The potentiometer appears near the bottom left corner of the image. The exact value of the potentiometer is not critical, but something in the range of 5–20 kOhms should allow for a range of spiking activity from the resonance oscillator as the potentiometer is adjusted. Note that the LED will blink every time a spike occurs, but the oscilloscope is used to visualize the time-dependent changes in voltage (across the LED).

Figure 1

A schematic diagram representing the electrical circuit from which the IF neuron equation is derived.

Figure 2

A graph showing the change in voltage over time (blue) of the circuit depicted in Figure 1 when current (0.1 nA, orange) is applied. X-axis is in ms, y-axis is in mV for membrane potential, 10⁻⁹ A for injected current. (See Appendix A for Python code.)

Figure 3

A graph showing a linear IF neuron based on the electrical equivalence circuit shown in Figure 1 and similar to the one shown in Figure 2, but with a “rule” imposed on the system that whenever membrane potential exceeds threshold (−35 mV in this case), the membrane potential suddenly (and inexplicably, based on the circuit in Figure1 and Equation 1–3) becomes −77 mV (approximately the Nernst equilibrium potential for potassium).

Figure 4

The electrical equivalence circuit for a nonlinear IF neuron with dynamics described by Equation 4–5. Note that resistors are now marked with associated conductances (g), and that the two batteries have opposite polarities, representing ENa (45 mV) and EL (−60 mV), respectively. (See Appendix B for photographs.)

Figure 5

The firing properties of the nonlinear IF neuron described in Figure 4 and by Equations 4–5. This model’s positive feedback makes it a truly spiking neuron model, although it still requires a voltage reset rule so that the action potential terminates in an after-hyperpolarization. The rule resets the voltage (again inexplicably according to the circuit in Figure 4 and Equations 4–5) to −77 mV when the membrane potential exceeds a threshold of +30 mV. [From Equation 5, h = 30 mV, s = 1 mV).]

Figure 6

The IV plot of the linear integrate-and-fire neuron described by Equation 1. Note that only one equilibrium exists (the x-intercept at the membrane’s resting leak potential), and that this equilibrium is stable (positive slope).

Figure 7

The IV plot of the nonlinear IF neuron described by Equations 4–5. The inset is included to show that a stable equilibrium still exists at the resting membrane potential (−60 mV). Note that there is also an unstable equilibrium at −37 mV and a pronounced region of negative slope resistance between −37 and −25 mV. There is also a second stable equilibrium at +45 mV (equilibrium potential for sodium). It is worth reminding the students that in a real cell, this stable equilibrium is ephemeral due to fast sodium channel inactivation and the opening of delayed-rectifier potassium channels.

Figure 8

A simple relaxation oscillator as an alternative electrical model of a spiking nerve cell. Note that the 2n2222 transistor is wired in a reverse-biased configuration, and that the base pin is intentionally floating (unconnected). V_out represents the point at which voltage is measured and fed to the data acquisition system or oscilloscope.

Figure 9

Spiking behavior of the relaxation oscillator. Upper trace shows the resting state, depolarization as current is applied by varying the potentiometer, the transition to a regenerative spiking state, and cessation of spiking upon lowering of the applied current back toward the minimum. Bottom trace shows near threshold spiking behavior as current is slowly lowered, with faster firing toward the left of the plot and a cessation of spiking just below threshold to the right. Note the subthreshold charging curve with measurable time constant.