Abstract#

We asked how the spike rate produced by a fully active pixel determines the average voltage feature passed to downstream classifiers. We compared direct membrane simulations with a simpler theory that treats the system as steady and approximately linear.

The theory reproduced the general filtering pattern and mean-response shape, but predicted responses that were too large and failed to capture variation between presentations. Accurately predicting voltage averaged over one presentation therefore still requires direct simulation of individual event counts and timings in this system.

Results#

Voltage moments across input rates#

Each presentation began from rest. We varied input rate and conductance per event, then averaged the voltage rise over the presentation (Fig. 1).

Higher rates should raise mean voltage. Variation could be nonmonotonic because event counts and timing differ between presentations and membrane responses are nonlinear.

Mean voltage rose with both input rate and event strength. Its SD rose at low rates, then levelled off or fell for the two weaker event strengths, but remained high for the strongest. This may reflect relatively steadier event counts plus shunting and saturation; we did not test these mechanisms separately.

Input-rate response distributions#

Here we kept presentation duration and event strength fixed, and changed only the input rate from sparse to dense drive (Fig. 2).

Input rate determines the expected number of events, not the exact number in any one presentation. A presentation may contain no events, one event or many; even equal event counts can give different average voltages when their timing differs.

At the lowest rate, most responses were near zero. The middle rate produced a mixture of sparse responses, while the highest rate produced a broad, almost continuous spread of responses.

Filtering before and after averaging#

This figure shows the theory’s response around three steady input rates. It is a theoretical prediction; we did not measure these curves by deliberately varying the input rate over time (Fig. 3).

The model responds similarly to slow changes, but increasingly suppresses changes that are faster than the synapse and membrane can follow. Stronger input also reduces the response to slow changes because it increases shunting and leaves less voltage difference to drive excitation. Averaging over a fixed window adds the regularly spaced dips and rebounds in the response.

Theory versus simulation#

The simpler theory replaces each random conductance history with its mean steady state, then assumes that the remaining fluctuations are small (Fig. 4).

If quantitatively accurate, this approximation would reproduce the simulated mean and SD curves.

Predicted means preserved the broad curvature and event-strength ordering (Pearson correlation 0.923), but were too large by a median factor of 1.975. The predicted and simulated SDs agreed poorly (Pearson correlation 0.291), especially for the largest event strength. The theory therefore explains the general filtering pattern, but not the amount of variation across finite presentations.

Methods#

Dataset#

Appendix: simulation and theory details#

Simulate one pixel’s voltage feature#

We set the normalized pixel intensity to its maximum, $𝑥=1$, and varied only its input rate ${𝑟}_{\text{input}}$, from 0 to 25 spikes/s. At each simulation step, the pixel either generated an event or did not. The event probability was

Here ${𝑟}_{\text{input}}$ is the input rate in spikes/s and $\mathrm{\Delta }{𝑡}_{\text{sim}}=0.1$ ms is the length of one simulation step. Over a presentation lasting ${𝑇}_{\text{present}}=200$ ms, the expected event count is ${𝑟}_{\text{input}}\frac{{𝑇}_{\text{present}}}{1000}$. We therefore controlled the rate; the number of events in an individual presentation remained random.

Each event increased AMPA conductance, which then decayed over time:

The indicator $𝑠\left[𝑘\right]$ equals one when step $𝑘$ contains an event and zero otherwise. The AMPA conductance at that step is $𝑔\left[𝑘\right]$; each event adds ${𝑤}_{\text{event}}$. Between events, conductance retains the fraction ${𝛽}_{\text{AMPA}}$ from the previous step, as set by the AMPA decay time ${𝜏}_{\text{AMPA}}$. Membrane voltage ${𝑉}_{𝑚\left(𝑡\right)}$ then follows Equation A4, where ${𝐶}_{𝑚}$ is membrane capacitance, ${𝑔}_{𝐿}$ is leak conductance, ${𝐸}_{𝐿}$ is the resting leak potential and ${𝐸}_{𝑒}$ is the excitatory reversal potential.

We used ${𝐶}_{𝑚}=1$ nF, ${𝑔}_{𝐿}=0.05$ μS, ${𝐸}_{𝐿}=-65$ mV, ${𝐸}_{𝑒}=0$ mV, ${𝜏}_{\text{AMPA}}=2$ ms, and event strengths ${𝑤}_{\text{event}}\in \left\{\text{0.6, 1.2, 2.4}\right\}$ μS. Every presentation began with no AMPA conductance, $𝑔\left(0\right)=0$, and the membrane at rest, ${𝑉}_{𝑚\left(0\right)}={𝐸}_{𝐿}$. We defined its voltage feature as

The feature ${𝑧}_{\text{feature}}$ is the average voltage above rest during the presentation. In the discrete simulation, we approximated this integral by averaging the voltage after each update and subtracting ${𝐸}_{𝐿}$.

For the mean and SD estimates in Figure 1, we generated 512 new presentations at each of 101 input rates and each conductance. For the full response distributions in Figure 2, we generated 4096 new presentations at each selected rate. This direct simulation preserves the random event counts and timings, the start from rest, and their combined effect within the finite presentation[1].

Replace random input with its steady average#

The simpler theory begins by replacing the changing AMPA conductance with its long-run average. For an ongoing Poisson input at rate $𝑟={𝑟}_{\text{input}}$, filtered-shot-noise theory[2] gives this mean as

Holding conductance at this mean gives the corresponding steady membrane voltage:

Because this voltage does not change during the presentation, the predicted mean feature is simply its rise above rest:

An overbar marks a steady average. Thus ${\overline{𝑔}}_{𝑟}$ is the mean conductance at input rate $𝑟$, ${\overline{𝑣}}_{𝑟}$ is the voltage produced by that mean conductance, and ${𝜇}_{\text{linear}}\left(𝑧\right)$ is the theory’s predicted mean voltage feature. From events to steady conductance and voltage derives Equations A6–A8 from the continuous conductance and membrane equations.

Predict responses to slow and fast changes#

Next, we asked how a small change in input rate would change voltage around the steady state. Treating these changes as small makes the equations linear and gives

Here $𝜔$ is angular frequency in rad/ms and $𝑖$ is the imaginary unit. The transfer function ${𝐺}_{𝑟\left(𝜔\right)}$ tells us how strongly voltage responds to an input change at each frequency. It describes small changes around the steady state, not the full start-from-rest simulation. Synapse and membrane response derives Equation A9.

The experiment does not use instantaneous voltage: it averages voltage over the presentation. That averaging has its own frequency response,

which multiplies the synapse-and-membrane response to give the complete response from input rate to voltage feature:

Here ${𝐴}_{{𝑇}_{\text{present}}}\left(𝜔\right)$ describes averaging over presentation duration ${𝑇}_{\text{present}}$, and ${𝐻}_{𝑟\left(𝜔\right)}$ describes the complete path from input-rate change to the averaged voltage feature. Averaging over the presentation derives Equations A10 and A11.

To generate Figure 3, we evaluated Equations A9 and A11 from 0.1 to 200 Hz at 0.25, 3, 25 spikes/s for the nominal 1.2 μS event strength. We converted frequency $𝑓$ in Hz to angular frequency using $𝜔=2𝜋\frac{𝑓}{1000}$. To compare response shapes, we expressed every magnitude relative to the steady response at the lowest input rate:

Here ${𝑀}_{𝑋\left(𝑓\right)}$ is response magnitude in decibels (dB). We used ${𝑋}_{𝑟}={𝐺}_{𝑟}$ for Figure 3A and ${𝑋}_{𝑟}={𝐻}_{𝑟}$ for Figure 3B. The variable $𝑓$ is the frequency of the input change in Hz, and ${𝑟}_{\text{low}}$ is the lowest steady input rate used as the reference.

Predict variation in the voltage feature#

An ideal Poisson event train has equal fluctuation power at every frequency. After subtracting its mean, its power across positive and negative frequencies on the millisecond time base is

The filter scales that input power by ${\left|{𝐻}_{𝑟\left(𝜔\right)}\right|}^2$. The predicted power spectrum of the voltage feature and its total variance are therefore

Here ${𝑆}_{\text{in}\left(𝜔\right)}$ is the input power at each frequency, ${𝑆}_{𝑧\left(𝜔\right)}$ is the resulting feature power, and ${\text{Var}}_{\text{linear}\left(𝑧\right)}$ is the predicted variance of the voltage feature. From Poisson events to feature variance derives Equations A13–A15.

We evaluated the integral in Equation A15 numerically on a logarithmically spaced frequency grid. We repeated the calculation with half as many points to check that the numerical result was stable.

Why the simpler theory misses the variation#

The simpler theory first averages conductance and then calculates the voltage response, written $𝑧\left(𝐸\left[𝑔\right]\right)$. The simulation instead calculates a response for every random conductance history and then averages those responses, written $𝐸\left[𝑧\left(𝑔\right)\right]$. Because voltage saturates, these two operations are not interchangeable. Here the approximate relationship is

The steady theory also treats input as if it acted throughout the whole presentation. In the simulation, an event arriving near the end contributes to the average in Equation A5 for only a short time.

More generally, the simulated feature distribution combines presentations with different event counts and, within each count, different event timings:

Here $𝑍$ is the random voltage feature, $𝑁$ is the event count, $𝑃\left(𝑁=𝑛\right)$ is the probability of observing $𝑛$ events, and ${𝑝}_{𝑍|𝑁}\left(𝑧|𝑛\right)$ is the distribution of feature values among presentations containing exactly $𝑛$ events.

Equation A15 instead describes small fluctuations around one steady state. At low rates, a presentation with no events stays exactly at rest, a single large event produces a response that depends strongly on its arrival time, and multiple events interact through shunting and voltage saturation. This irregular mixture explains why the theoretical SD rose too sharply and peaked too early, especially for the strongest events.

Appendix: how the simpler theory is derived#

From events to steady conductance and voltage#

In the simulation, $𝑠\left[𝑘\right]$ records whether step $𝑘$ contains an event. For the continuous-time calculation, we represent the same event sequence as $𝑠\left(𝑡\right)={\sum }_{𝑗}𝛿\left(𝑡-{𝑡}_{𝑗}\right)$: a train of instantaneous impulses at event times ${𝑡}_{𝑗}$. AMPA conductance then follows

Here $𝛿\left(𝑡-{𝑡}_{𝑗}\right)$ is a Dirac impulse marking event $𝑗$ at time ${𝑡}_{𝑗}$. Each impulse adds ${𝑤}_{\text{event}}$ conductance, while existing conductance decays with time constant ${𝜏}_{\text{AMPA}}$.

A steady input rate of $𝑟$ spikes/s is $\frac{𝑟}{1000}$ events per millisecond. To find the long-run mean conductance, we average Equation B1 and set its mean rate of change to zero:

Solving this equation gives Equation A6. At the corresponding steady membrane voltage, inward and outward current balance:

Rearranging the terms containing ${\overline{𝑣}}_{𝑟}$ gives Equation A7.

Small changes through the synapse and membrane#

We write each changing quantity as its steady value plus a small fluctuation:

Substituting these expressions into Equation B1 makes the steady terms cancel, leaving an equation for the conductance fluctuation:

In the frequency domain, differentiation with respect to time becomes multiplication by $𝑖𝜔$. The conductance response is therefore

We apply the same decomposition to the membrane equation, Equation A4. The steady terms cancel through Equation B2. Because the theory assumes small fluctuations, it drops their product $𝛿𝑔𝛿𝑣$, which is second order. This leaves

Transforming Equation B5 into the frequency domain and substituting Equation B4 gives Equation A9. Its denominator contains two filters: one from AMPA conductance and one from the membrane. The numerator contains the conductance added per event and the local voltage difference driving excitation.

Averaging over the presentation#

After assuming that fluctuations are small, the feature in Equation A5 is simply their average voltage during the presentation:

Averaging gives equal weight, $\frac{1}{{𝑇}_{\text{present}}}$, to every time from zero to ${𝑇}_{\text{present}}$ and zero weight outside that interval. The frequency response of this rectangular averaging window is

which is Equation A10. Because the averaging happens after the membrane response, their two frequency responses multiply, giving Equation A11. The identity $1-exp\left(-𝑖𝑥\right)=2𝑖exp\left(-𝑖\frac{𝑥}{2}\right)sin\left(\frac{𝑥}{2}\right)$ shows that the magnitude is

With ${𝑇}_{𝑠}=\frac{{𝑇}_{\text{present}}}{1000}$ s and $𝜔=2𝜋\frac{𝑓}{1000}$,

This response becomes zero at $𝑓=\frac{𝑛}{{𝑇}_{𝑠}}$, where $𝑛$ is any nonzero integer. At those frequencies, the presentation contains a whole number of cycles, so positive and negative parts cancel in the average. The repeated dips and rebounds come from this sinc-shaped averaging response; the synapse and membrane make the overall response fall at higher frequencies.

From Poisson events to feature variance#

For ideal Poisson input at $\frac{{𝑟}_{\text{input}}}{1000}$ events per millisecond, counts in separate time intervals are independent and the variance of each count equals its mean. After subtracting the mean event rate, the input autocovariance is

Here ${𝐶}_{𝛿𝑠}\left(\ell \right)$ measures how the centred input relates to itself after a time lag $\ell$. The Dirac delta $𝛿\left(\ell \right)$ means that this ideal input is correlated with itself only at zero lag.

The frequency transform of a Dirac delta is constant, which gives the flat input spectrum in Equation A13. A linear filter multiplies that spectrum by the squared magnitude of its frequency response, giving Equation A14. Adding the feature power across all frequencies then gives its variance:

Substituting Equation A14 gives Equation A15. This result is exact for the simplified model with steady input and small linear fluctuations. It is only an approximation to the simulated system, which is nonlinear, begins each presentation from rest and contains a finite random number of events.

References#

1. Marco Brigham and Alain Destexhe: Nonstationary Filtered Shot-Noise Processes and Applications to Neuronal Membranes. Physical Review E, 2015. doi:10.1103/PhysRevE.91.062102
2. Lars Wolff and Benjamin Lindner: Mean, Variance, and Autocorrelation of Subthreshold Potential Fluctuations Driven by Filtered Conductance Shot Noise. Neural Computation, 2010. doi:10.1162/neco.2009.02-09-958