Mathematics of GT models

Introduction

Suppose our network has n neurons N. We want to run a simulation of the membrane potential v(t) of each neuron N over time t. Naturally, the system evolves so that: v(t+1) = v(t) + \left.\frac{\partial v}{\partial t}\right|_{t+1} The hard part, of course, is to compute \partial v/\partial t. That is, how do membrane potentials evolve over time?

Power minimization

To tackle this question, Gangopadhyay and Chakrabartty applied the Baum-Eagon inequality to the net metabolic power \mathcal{H} of v to derive \partial v/\partial t as a growth-transform of v. That is, they assumed that \partial v/\partial t is always such that: \mathcal{H}(v)\leq \mathcal{H}(v + \partial v/\partial t) This amounts to assuming that \mathcal{H} is minimized over time: \frac{\partial\mathcal{H}}{\partial v}\frac{\partial v}{\partial t} \leq 0 Here, we will work directly from this minimization assumption and use a biologically inspired interpretation of the variables to derive \partial v/\partial t. For simplicity, let us assume that \partial\mathcal{H}/\partial t = 0, i.e., that the metabolic power is not changing over time.

Also, let us assume that v represents only the subthreshold membrane potential and thus that \partial v/\partial t only directly models subthreshold membrane potential dynamics.¹ This leaves out any mechanistic modeling of spikes themselves. Instead, spikes are modeled as a constant and instantaneous value v_\mathrm{spike} (variable spike_potential in the simulation) added to the subthreshold dynamics when the spike threshold is crossed. We will explain below how spikes are nevertheless still critical to the evolution of the system.

Where do we begin? It’s plausible that the membrane potential v is a scalar multiple fv_\mathrm{rest} of the resting potential v_\mathrm{rest},² where the scale factor f is a function of \partial\mathcal{H}/\partial v. In other words, we assume that the scale factor f depends on how a small change \partial v in membrane potential would change net metabolic power \mathcal{H}. Hence, we have that:³ v(t+1) = v(t) + \left.\frac{\partial v}{\partial t}\right|_{t+1} = f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_{t+1}\right)v_\mathrm{rest}

Scale factor

Therefore, if we could find a suitable scale factor f, we’d be able to compute \partial v/\partial t as: \left.\frac{\partial v}{\partial t}\right|_t = f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_{t}\right)v_\mathrm{rest} - v(t-1) Combining the previous equation with the minimization assumption, we have: \frac{\partial\mathcal{H}}{\partial v} \left(f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t\right)v_\mathrm{rest} - v(t-1) \right) = 0 Assuming \partial\mathcal{H}/\partial v \neq 0, this equation implies that: v(t-1) = f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t\right)v_\mathrm{rest} And so: f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t\right) = \frac{v(t-1)}{v_\mathrm{rest}} Hence, we see that f is a function of the metabolic power gradient \partial\mathcal{H}/\partial v that’s equal to the ratio of v over the rest potential v_\mathrm{rest}. An immediate consequence is that, when v\approx v_\mathrm{rest}, the scale factor f is approximately 1. Hence, we are assuming that v_\mathrm{rest} is an equilibrium point (as defined by dynamical systems theory) of the membrane potential.

Notice that this derivation implies that \partial v/\partial t=0, which may seem obviously wrong. However, we are attempting to derive a formula for \partial v/\partial t which makes \partial\mathcal{H}/\partial t=0. The “makes” is aspirational: of course, \partial v/\partial t\neq 0, but, what general form of \partial v/\partial t will tend to make \partial\mathcal{H}/\partial t=0 over time? In terms of biology, we can alternatively think of the system as trying to maintain a stable rest voltage v_\mathrm{rest}, and the question is: what form of \partial v/\partial t will best achieve this goal?

Rest vs spike power

In order to solve for f, it’s helpful to think about the metabolic power gradient \partial\mathcal{H}/\partial v and how it relates to two quantities: the metabolic power \mathcal{H}_\mathrm{rest} of maintaining rest potential v_\mathrm{rest} and the metabolic power \mathcal{H}_\mathrm{spike} of initiating a spike under the conditions which hold at time t. As a linear approximation, we have that: \mathcal{H}_\mathrm{rest} = v_\mathrm{rest}\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t This equation is justified because the metabolic power \mathcal{H}_\mathrm{rest} of maintaining rest potential v_\mathrm{rest} is equal to the power used to maintain the negative potential v_\mathrm{rest} under the natural positive flow of charge into the cell. Unit analysis (\mathrm{Watts} = \mathrm{Volts} \times \mathrm{Ampere}) implies that this latter quantity must be the power gradient \partial\mathcal{H}/\partial v evaluated at t.

What about the metabolic power \mathcal{H}_\mathrm{spike} of initiating a spike? If v_\mathrm{threshold} is the membrane potential at spike threshold, then (by the above unit analysis) the change \partial I_\mathrm{influx} in current I across the membrane at spike initiation is: \partial I_\mathrm{influx} = \left.\frac{\partial\mathcal{H}}{\partial v}\right|_{v_\mathrm{threshold}} and the metabolic power required to produce a spike is approximately: \mathcal{H}_\mathrm{spike} = v\partial I_\mathrm{influx}

The values \mathcal{H}_\mathrm{rest} and \mathcal{H}_\mathrm{spike} are helpful because, at any moment t, whether there is a spike depends on how they relate. If \mathcal{H}_\mathrm{spike} < \mathcal{H}_\mathrm{rest}, then the neuron will be more inclined to spike. Under this assumption, we can suppose that f is a function of the difference between \mathcal{H}_\mathrm{spike} and \mathcal{H}_\mathrm{rest}: \begin{aligned} f\left(\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t\right) &= f\left(\mathcal{H}_\mathrm{spike} - \mathcal{H}_\mathrm{rest}\right) \\ &= f\left(v\partial I_\mathrm{influx} - v_\mathrm{rest}\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t\right) \end{aligned} A suitable function f follows from the previously cited lemma that f\approx 1 when v\approx v_\mathrm{rest}. A plausible way to achieve this result without f collapsing into a trivial constant is to set: f = \frac{ v\partial I_\mathrm{influx} - v_\mathrm{rest}\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t }{ v_\mathrm{rest}\partial I_\mathrm{influx} - v\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t }

Interpretation

While the motivation for this definition of f is mathematical (we want f\approx 1 when v\approx v_\mathrm{rest}), the equation makes some biological sense. We have already discussed the numerator: v_\mathrm{rest}\left.\partial\mathcal{H}/\partial v\right|_t is the metabolic power \mathcal{H}_\mathrm{rest} of maintaining rest potential v_\mathrm{rest} under current influx at time t, v\partial I_\mathrm{influx} is the metabolic power \mathcal{H}_\mathrm{spike} of initiating a spike under that same condition, and their difference signals whether it takes more metabolic power to spike or maintain rest.

In the denominator, v_\mathrm{rest}\partial I_\mathrm{influx} is the metabolic power of initiating a spike from the rest potential, while v\left.\partial\mathcal{H}/\partial v\right|_t is the metabolic power of maintaining the potential v(t) under the current influx \left.\partial\mathcal{H}/\partial v\right|_t holding at time t. Hence, the difference between these two quantities is the amount of additional metabolic power needed to (hypothetically) initiate a spike from rest, compared to the power needed to maintain the cell’s present state.

Thus, the denominator provides a kind of upper bound, or normalization. The scale factor f is plausibly interpreted as the additional power needed to spike (vs maintain rest potential) as a fraction of the max possible power needed to initiate a spike. It gives a normalized “cost” of the power (and hence energy) for a spike.

Power gradient

Returning to \partial v/\partial t, we have: \left.\frac{\partial v}{\partial t}\right|_t = v_\mathrm{rest} \left( \frac{ v\partial I_\mathrm{influx} - v_\mathrm{rest}\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t }{ v_\mathrm{rest}\partial I_\mathrm{influx} - v\left.\frac{\partial\mathcal{H}}{\partial v}\right|_t }\right) - v(t-1) This is the equation given by Gangopadhyay and Chakrabartty for GT models. It ensures the minimization condition, that is, ensures that \partial\mathcal{H}/\partial t = 0.

How do we determine \partial I_\mathrm{influx}? Well, \partial I_\mathrm{influx} is the change in current I across the membrane at spike initiation. Following Gangopadhyay and Chakrabartty, we will treat it as a constant empirical parameter that bounds the power gradient, i.e., \partial I_\mathrm{influx} \geq \left.\partial\mathcal{H}/\partial v\right|_t for all t. This treatment is plausible, as presumably the moment of spike initiation involves the largest change in membrane current a neuron will experience.⁴

Second, how do we determine \partial\mathcal{H}/\partial v? Intuitively (and, as before, following Gangopadhyay and Chakrabartty), we have: \frac{\partial\mathcal{H}}{\partial v} = I_\mathrm{synaptic\;transmission} - I_\mathrm{membrane\;current} + I_\mathrm{spike} In this equation, I_\mathrm{synaptic\;transmission} is the input current induced by synaptic transmission across all synapses, I_\mathrm{membrane\;current} gives the current across the membrane of each neuron due to external stimuli (I_\mathrm{stim}) and intrinsic membrane leak (I_\mathrm{leak}), and I_\mathrm{spike} gives the spike current (if any). The stimulus current I_\mathrm{stim} is what’s specified by the simulation variable stimulus_current_matrix, while the membrane leak current I_\mathrm{leak} is determiend by the cell type parameter leak_conductance. The spike current I_\mathrm{spike} is determined by simple thresholding: I_\mathrm{spike}=0 if v < v_\mathrm{threshold} and otherwise is equal to the value of the simulation variable I_spike for the cell type.

The input current induced by synaptic transmission across all synapses, I_\mathrm{synaptic\;transmission}, is handled specially to account for signal transmission lag, as described in the tutorial on spatial GT models. However, the basic idea is straightforward: the synaptic current is the synaptic conductance times the presynaptic membrane potential v.

Spiking without spikes?

GT models are only ever modelling subthreshold dynamics. That is, v represents only the membrane potential below the spike threshold. This arrangement leads to an awkward interpretation of I_\mathrm{synaptic\;transmission}. Specifically, computing I_\mathrm{synaptic\;transmission} as synaptic conductance times v implies that it’s the subthreshold membrane potential of the presynaptic neuron which is transduced across the synapse to the postsynaptic neuron. Biologically, signal transduction at synapses happens primarily through spikes, of course. So, two questions naturally arise:

If it’s the subthreshold membrane potential v which drives synaptic transmission in a GT model, how do the spikes themselves have any causal role in signal transmission?
Whatever the answer to the first question, why arrange a spiking neural network model in this strange way?

The answer to the first question is that, in GT models, spikes of presynaptic neurons causally affect the membrane potential of postsynaptic neurons indirectly, via their effect on the subthreshold membrane potential of the presynaptic neuron. This is, of course, the inverse of the causal flow in the real biology. In real neurons, changes in subthreshold membrane potential cause spikes, which cause synaptic transmission. In GT models, the energy cost of a future spike causes changes in the subthreshold membrane potential, which itself is engaged in continuous synaptic transmission. Mathematically, that future energy cost appears as the term I_\mathrm{spike} in the metabolic power gradient \partial\mathcal{H}/\partial v, and as the term \mathcal{H}_\mathrm{spike} in the scale factor f.

As for the second question, the advantage of flipping the causal order and modelling the energy cost of a spike instead of a spike itself is that it makes it easier to optimize the synaptic “weights”, i.e., the matrix Q of synaptic conductances, so that the network model reproduces a desired response in v for a given stimulus input current I_\mathrm{stimulus\;input}. Traditionally, spiking neural networks are trained by optimizing synaptic weights to minimize the difference between some time-dependent desired spike rate and the model spike rate. However, if R(w,t,n) is the spike rate implied by a spiking neural network for neuron n at time t under synaptic weights w, R will not usually be differentiable due to spiking being a function of discrete threshold crossings. This means that spiking neural networks can’t straightforwardly be trained using gradient descent. In contrast, \mathcal{H} is a continuous function of v with a well defined derivative \partial\mathcal{H}/\partial v, and v itself, being only the subthreshold membrane potential, is also smooth.

Membrane potential reset

Notice that unlike a leaky integrate-and-fire neuron, \partial v/\partial t itself produces the reset after a spike. When v is very near the threshold v_\mathrm{threshold}, then (by the definition given above) we have that: \partial I_\mathrm{influx} = \left.\frac{\partial\mathcal{H}}{\partial v} \right|_{v_\mathrm{threshold}} and hence f\approx -1, because, in general, (x-y)/(y-x)=-1. Thus, when v\approx v_\mathrm{threshold}: v + \frac{\partial v}{\partial t }\approx -v_\mathrm{rest} At first glance, this seems off by a sign: shouldn’t $v/t $ return us to v_\mathrm{rest} when near v_\mathrm{threshold}? Yes, but the issue is easy enough to fix: simply add a negation to our definition of f. In the actual DACx code, the relevant multiple for f is not v_\mathrm{rest}, but |v_\mathrm{rest}| + \epsilon for some small \epsilon, but the result is the same: the model produces a reset to a value near the rest potential after a spike, without any explicit reset mechanism.

Michael Barkasi

August 1, 2026 (v1.0)