https://orcid.org/0000-0003-0702-792X
This is an uncorrected proof.
Figures
Abstract
In sensory systems, stimuli are represented through the diverse firing responses and receptive fields of neurons. These features emerge from the interaction between excitatory (E) and inhibitory (I) neuron populations within the network. Changes in sensory inputs alter this balance, leading to shifts in firing patterns and the input-output properties of individual neurons and the network. Although these phenomena have been extensively investigated experimentally and theoretically, the principles governing how E and I inputs are integrated remain unclear. Here, probabilistic rules are derived to describe how neurons in feedforward inhibitory circuits combine these inputs to generate stimulus-evoked responses. This simple model is broadly applicable, capturing a wide range of response features that would otherwise require multiple separate models, and offers insights into the cellular and network mechanisms influencing the input-output properties of neurons, gain modulation, and the emergence of diverse temporal firing patterns.
Author summary
Sensory stimuli activate networks of excitatory and inhibitory neurons whose interactions shape how the brain represents information. An individual neuron’s response therefore depends not only on the strength of excitatory input, but also on how inhibition is recruited as stimulus conditions change. These interactions alter firing thresholds, response gain, and temporal firing patterns, yet the principles governing how excitatory and inhibitory inputs combine remain unclear. In this study, I develop a simple probabilistic framework to describe how excitatory and inhibitory synaptic inputs interact in feedforward inhibitory circuits. I express neuronal input–output relationships in terms of the probability that excitation survives coincident inhibition, thereby linking firing responses directly to identifiable synaptic and network parameters. Using this framework, I show that the model accounts for key features observed in sensory systems, including multiplicative and additive gain modulation, non-monotonic input–output curves, and diverse temporal firing patterns evoked by brief or sustained stimuli. By unifying these phenomena within a single, analytically tractable description, I provide insight into how changes in excitatory–inhibitory balance flexibly regulate neuronal responses across sensory conditions and behavioral states.
Citation: Reyes AD (2026) Computing the effects of excitatory-inhibitory balance on neuronal input-output properties. PLoS Comput Biol 22(3): e1013958. https://doi.org/10.1371/journal.pcbi.1013958
Editor: Sacha Jennifer van Albada, Research Centre Jülich: Forschungszentrum Jülich GmbH, GERMANY
Received: May 16, 2025; Accepted: January 29, 2026; Published: March 9, 2026
Copyright: © 2026 Alex D. Reyes. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The source code and data used to produce the results and analyses presented in this manuscript are available on a Github repository at https://github.com/AlexDReyes/ReyesPlosCompBio2025.git.
Funding: A.R. was supported by grant 1 R01 MH129031-01 National Institutes of Health https://www.nih.gov/. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The author has declared that no competing interests exist.
Introduction
A common motif in sensory systems is the feedforward inhibitory circuit, where excitatory afferents from an external source synapse onto both excitatory (E) and inhibitory (I) neurons, with inhibitory neurons then synapsing back onto E cells [1–5]. During stimulation, interactions within this circuit generate complex dynamics and shape the receptive field properties of neurons. As stimulus parameters vary, the balance between E and I inputs shifts, leading to both qualitative and quantitative changes in neuronal sensitivity and evoked firing patterns [1,4,6,7].
A neuron’s response to a stimulus is characterized by its input–output (I–O) curve, where the input typically refers to synaptic current and the output to firing rate. Compared to neurons in vitro [8,9], neurons in vivo exhibit more diverse I–O profiles. Background excitation and inhibition from ongoing network activity introduce membrane potential fluctuations, allowing neurons to respond even to weak inputs that would otherwise remain subthreshold, and to generate smoothly increasing I–O curves [10–13]. In some cases, responses initially increase with stimulus intensity but then decline after reaching a peak [14–17], likely due to strong inhibitory recruitment [18,19]. I–O curves also depend on stimulus duration: brief stimuli (5–50 ms) evoke responses sensitive to the timing of excitation and inhibition [2,4,20,21], while longer stimuli (hundreds of milliseconds) evoke responses that vary with the average synaptic current generated during barrages [22–24].
Moreover, the I–O curves may change depending on the animal’s state, such as when it is at rest, in motion [25,26], or actively attending [27,28]. To maintain selectivity to stimulus features across states, the slope of the I–O curve should change without affecting the minimum input needed to evoke firing [13,29]. This multiplicative (or divisive) gain modulation can arise from feedback from neighboring E and/or I neurons [30–34], feedforward inhibition [21], or the combination of synaptic noise and conductance [10]. Additive (or subtractive) modulation, by contrast, shifts the activation threshold without altering gain, thereby affecting tuning curve width [13].
Finally, differences in E–I balance can produce diverse temporal firing responses. Some neurons exhibit continuous firing throughout the stimulus duration, while others fire transiently at the stimulus onset [16,22,35–39] and/or at the offset [39–42]. These firing profiles are observed in cortical and subcortical neurons [42] and may be generated locally [22,41–44] or inherited from upstream sources [40]. Additionally, a neuron’s response type may change based on the stimulus intensity [16] or whether the preferred stimulus is presented [35,38,45,46].
Identifying common operating principles across these phenomena will provide valuable insights into potential mechanisms. This study aims to derive rules for calculating E–I balance in feedforward inhibitory circuits. The model combines E and I inputs probabilistically and links the associated changes in responses and I–O curves to the synaptic and network properties. The model reproduces and clarifies the conditions for gain modulation, non-monotonic I–O curves, and diverse firing patterns.
Results
The following sections begin with an idealized feedforward inhibitory network to introduce core concepts of the probabilistic interaction between excitatory and inhibitory inputs. These principles are then extended to more physiologically realistic conditions that include multiple, temporally distributed inputs. Finally, the model is applied to examine neuronal input–output properties, such as gain modulation and temporal firing profiles.
Simple model
Consider a hypothetical circuit consisting of a postsynaptic excitatory neuron (henceforth termed the “reference cell”) and a single inhibitory neuron, both receiving an excitatory postsynaptic potential (EPSP) from a common external afferent (Fig 1A). During stimulation, the afferent evokes an EPSP in both cells with probability , where
denotes the probability that an EPSP occurs in a given target cell.
A, top, schematic of a simple feedforward inhibitory circuit. A postsynaptic excitatory neuron (black triangle) and a local inhibitory interneuron (red circle) both receive a single excitatory input from an external afferent. An IPSP is generated in the postsynaptic E neuron whenever the interneuron fires. Bottom, EPSPs (blue) and IPSPs (red) recorded in the postsynaptic E neuron over seven stimulus trials. An EPSP is shown as canceled when it coincides with an IPSP, but more generally Eq 1 describes the probabilistic outcome: over n trials, the expected number of uncanceled EPSPs is . B, surface plot of the net probability as a function of and
. Orange curves show predicted firing probability when increases linearly with (
) at different scaling factors
. Green curves correspond to cases where is held constant. Inset, projection of these curves onto the – plane. If each EPSP is suprathreshold, is equivalent to the firing probability
, and the plot can be interpreted as the neuron’s input–output relation.
The firing of the I cell—and consequently the occurrence of an inhibitory postsynaptic potential (IPSP) in the reference cell—requires that an EPSP first arise in the I neuron. Conditional on the occurrence of an EPSP in the I neuron, that neuron fires with probability
Thus, the joint probability that an EPSP appears in the I neuron and that the I neuron subsequently fires to generate an IPSP in the reference cell is .
Fig 1A shows four EPSPs (blue) and four IPSPs (red) across seven stimulus trials. For illustrative purposes, EPSPs and IPSPs are assumed to have the same amplitude and latency. An EPSP that coincides with an IPSP is effectively canceled (trials 3 and 6), whereas trials in which the IPSP fails to appear (trials 1 and 4) or appears alone (trials 2 and 7) have no effect. Thus, the probability that an EPSP is not canceled by an IPSP is
Eq 1 should be interpreted as a trial-averaged survival probability, reflecting both the chance that an excitatory input occurs () and the chance that it is not canceled by coincident inhibition (
). Over n independent trials, the number of uncanceled EPSPs in the reference cell follows a binomial distribution with mean
.
The surface plot in Fig 1B illustrates how varies with
and
, highlighting key properties of Eq 1. First,
is nonzero even when
, except in the limiting cases
or
. Second,
depends on how
and
co-vary. For example, if
scales with
as
with
, then
increases monotonically with
for small
, but becomes non-monotonic for larger
(orange curves). Third, if
is fixed,
increases linearly with
at a rate given by the slope
(green curves).
The inset shows the orange and green curves projected onto the –
plane. If each EPSP is suprathreshold,
corresponds to the firing probability
, and the resulting plot represents the I–O relation of the neuron, with
serving as a proxy for stimulus intensity or feature. This framework thus links the probabilistic structure of synaptic inputs to the macroscopic I–O behavior of the neuron.
In the following sections, this toy model is extended to more realistic networks with multiple afferents and inhibitory neurons, where cancellation emerges statistically.
General model
Neuronal responses depend in part on the duration of stimulation. During sustained input, afferents and inhibitory neurons generate sequences of excitatory and inhibitory synaptic inputs, respectively, producing a synaptic barrage in the postsynaptic reference neuron. If the input is sufficiently strong, neurons fire repetitively at a rate determined by the average synaptic current [23]. In contrast, brief stimuli evoke EPSPs and IPSPs that arrive in close temporal proximity, placing the neuron in a regime where spiking is highly sensitive to both the amplitude and timing of synaptic inputs [4]. These two regimes will be treated separately, as the model yields distinct predictions for each. The case with sustained stimulation is examined first.
Sustained stimuli: Oscillatory firing regime.
The response of the reference cell to long-duration stimuli was evaluated in three stages (Fig 2A). First, Poisson-distributed spike trains from external afferents (blue) were generated to produce excitatory synaptic barrages, which were delivered to both the reference cell and the I neurons, modeled as leaky integrate-and-fire (LIF) units. Second, the spike trains evoked in a specified number of I neurons (red) were summed to form the inhibitory synaptic barrage. Finally, the excitatory and inhibitory barrages were combined and delivered to the reference cell, and its firing response was measured across a range of conditions.
A, Schematic of the network consisting of a reference cell (R, triangle) and inhibitory neurons (red circles). During stimulation, each of the
afferents fires with probability
at rate
for a duration of 0.1 to 1 second. Each afferent spike evokes an EPSC with amplitude
in both the reference neuron and each inhibitory neuron with probabilities
and
, respectively. If the excitatory synaptic barrage was sufficiently large, each inhibitory neuron fired with probability
at a rate
. The effective probability—which factors in network variables, synaptic strength, and firing rates—that the spikes reach the reference cell is
(see text). Finally, the probability that each inhibitory spike evoked an inhibitory postsynaptic current (IPSC) with amplitude
in the reference neuron is
. In the simulations,
and
are set to 1. Bi, top, Time course of excitatory synaptic current barrage (blue, left ordinate) when
(right ordinate) was ramped to a steady-state value of 0.35. The inhibitory barrage (red), plotted as an absolute value for comparison, develops after a short delay of
ms, eventually reaching a comparable steady-state value. The resulting net current (black), calculated with probability
, shows a transient peak at stimulus onset followed by a lower steady-state level. In contrast, the unconditioned current obtained by direct subtraction of excitatory and inhibitory currents is nearly zero (gray). Bottom, Spike histogram of the reference neuron. Inset, magnified view of the tonic firing component. ii, iii, Same as i, except with
and
set to 0.55 and 0.85, respectively. Histograms compiled with bin width
ms and with 10,000 sweeps. Model parameters, Simulations used leaky integrate-and-fire neurons (see Methods) with
,
,
Hz, and
Hz. EPSC and IPSC are alpha functions with amplitudes
and
(pA). The steady-state value of
was controlled by fixing the input probability to inhibitory cells (
in Eq 2) at 0.35 and adjusting the ratio
(Eq 3) to 0.3 (i), 0.4 (ii), and 0.7 (iii). Bin width
ms.
To introduce the key variables manipulated in the simulations, the relationships between the E and I probabilities and the corresponding synaptic currents are first developed. A detailed description of the model equations and parameters is provided in the Methods and S1 Appendix; here, only the principal features necessary for interpreting the Results are summarized.
Fig 2A shows a schematic of the feedforward network, with the parameters defined in Table 1. During a prolonged stimulus, each afferent generated a Poisson train of action potentials with mean rate , where
is the probability that an afferent becomes active during the stimulus,
is the probability that each spike produces an excitatory postsynaptic current (EPSC) in the reference cell, and
is the firing rate conditional on activation. Across afferents or repeated trials, this product represents the effective EPSC rate, and the probability of observing a spike within a time bin of width
is approximately
. In this formulation,
captures variability across afferents or stimulus presentations,
quantifies synaptic efficacy, and
describes the within-trial firing dynamics of an active afferent.
When the afferents fired, each spike evoked an EPSC, whose integral yielded the total charge transfer
. The mean steady-state excitatory currents to the reference neuron (
) and to the inhibitory neurons (
) are shown in Fig 2B (blue; see S1 Appendix). When the afferent input was sufficiently large, the I neurons fired at a rate
and generated inhibitory synaptic current (
) in the reference cell. These mean currents are expressed as
Here, denotes the probability that an afferent spike evokes an EPSC in an inhibitory neuron, and
denotes the probability that an inhibitory spike evokes an IPSC in the reference cell [8,47–50]. For convenience, we define the effective excitatory drive to inhibitory neurons as
.
The term denotes the effective probability that an inhibitory afferent to the reference cell fires during stimulation, or equivalently, the probability that an inhibitory spike reaches the reference cell. It is expressed as a product of factors that relate inhibitory neuron number, synaptic strength, and activity level to those of the excitatory afferents (see S1 Appendix):
where is the probability that an inhibitory neuron fires during the stimulus,
is the inhibitory-to-excitatory input ratio,
is the ratio of unitary charge transfers,
is the inhibitory-to-excitatory rate ratio, and
is the synaptic efficacy ratio. By construction,
incorporates these relative differences in number, strength, and activity, allowing the inhibitory current (Eq 2) to be written in terms of
,
,
, and
.
The mean net current to the reference cell, under the condition that inhibition is contingent on coincident excitation, is given by (see also S12 Eq in S1 Appendix)
where denotes the survival probability of excitation—requiring both that an excitatory input occurs with probability
and that it is not canceled by coincident inhibition with probability
(see S1 Appendix for details). This compact probabilistic form incorporates interneuron recruitment and synaptic reliability, and provides an effective representation of the mean net excitatory drive under feed-forward coupling. In contrast to classical add–subtract models (e.g., for Poisson processes [51]), in which excitation and inhibition are treated as independent and subtracted directly, the present formulation yields a probabilistic measure that, by construction, remains within a normalized range (approximately
) for realistic values of identifiable parameters (see Discussion), thereby avoiding the unbounded subtraction of independent terms.
Fig 2Bi shows the synaptic currents (top) and spike histograms (bottom) recorded from the reference cell during stimulation. Simulations were performed using LIF neurons (see figure caption and Methods for details). The afferent input probability was ramped over time to a steady-state value of , producing a barrage of excitatory synaptic currents (blue). Physiologically, this ramp mimics the gradual increase in stimulus intensity before reaching a plateau. When inhibition was modeled as independent of coincident excitation, inhibitory currents (red) were recruited after a short delay and rose to a comparable level. In this case, the unconditioned net current—obtained by direct subtraction of excitatory and inhibitory currents,
(gray)—was nearly zero, and no firing occurred. By contrast, when inhibition was conditioned on coincident excitation, the net synaptic current (black, top) and the firing profile of the reference cell (bottom) displayed a small transient peak followed by a sustained tonic component.
The net current and tonic firing reflected the balance between and
, in agreement with the toy model (orange curves in Fig 1B). When both probabilities increased together,
and the tonic firing rate rose to a peak value (Fig 2Bii) before declining (iii), and eventually vanished as
and
approached 1.
Simulations were performed while systematically increasing . The resulting input–output (I–O) curve was obtained by plotting the evoked firing rate (mean ± SD) against
(Fig 3A). When the mean net input current
exceeded the rheobase
, the reference neuron entered the oscillatory firing regime. In this regime, the average firing rate of the LIF model (green curve) followed the standard analytical solution (see S16 Eq of S1 Appendix).
A, Representative input–output (I–O) curve showing mean firing rate (mean ± SD) as a function of, with (
,
,
). Superimposed are the predicted firing rates in the sub-oscillatory (cyan) and oscillatory (green) regimes. Inset, Example membrane potential trace evoked by a sub-rheobase input. Fluctuations in the membrane potential cause threshold crossings (orange line). B, I–O curves for different fixed values of
. The ratio
in Eq 3 was varied by fixing
at specified levels (range: 0–0.8), which in turn modulated
. Inset, Plots of
,
, and
versus
. C, Same as B, except
increased linearly with
, implemented by setting
while holding
,
, and
(see text). D, Same as C, except
was allowed to vary with for different values of
. Inset,
remained near zero for low
and increased linearly with
once threshold was crossed. Solid and dotted curves correspond to
and 0.4, respectively.
Firing could still occur with weak inputs due to voltage fluctuations that occasionally crossed threshold, even when was below rheobase (Fig 3A, inset). In this fluctuation-dominated (sub-oscillatory) regime (cyan curves), the firing rate was given by
where is the minimum oscillatory firing rate at rheobase and
is the probability that the excitatory drive exceeds threshold (derivation in S1 Appendix). The resulting
increased sigmoidally with
and approached a maximum near the onset of the oscillatory regime. Across the full range of
, the overall firing rate was given by the larger of
and
.
The model predicts that, for a fixed , the slope of the I–O curve scales in proportion to
(Fig 1B, green; Eq 1). Consistent with this prediction, increasing
(see figure caption for details) caused the I–O curves to exhibit a reduction in slope. However, there was also a rightward shift, reflecting a higher activation threshold (Fig 3B). This condition, in which inhibition was held constant across the full range of
, is analogous to experiments in which inhibitory neurons are continuously activated optogenetically during sensory stimulation [34]. The slope decrease corresponds to multiplicative gain modulation, whereas the threshold shift reflects an additive effect of persistent inhibition, requiring stronger excitation to elicit firing. These effects were observed in both the oscillatory and fluctuation-dominated regimes and were captured by the analytical expressions for
(green) and
(cyan). Together, these results demonstrate that inhibition modulates the I–O relationship through a combination of multiplicative and additive mechanisms [13,29].
The I–O curves exhibited either monotonic or non-monotonic increases with , depending on how strongly inhibition co-varied with excitation (Fig 3C), consistent with model predictions (Fig 1B, orange curves). Co-variation of
with
was implemented by allowing
to increase linearly with
(inset). Physiologically, this corresponds to the progressive recruitment of inhibitory neurons with increasing stimulus intensity. For small
, firing rates increased monotonically with
(Fig 3C), whereas as
approached 1, the I–O curves flattened and eventually became non-monotonic. Notably, the minimum
required to evoke firing (
) remained unchanged, while the slope of the rising phase decreased. Thus, when
was restricted to the range in which the I–O curve increased, gain modulation was effectively multiplicative.
Finally, the model predicts changes in the I–O curves of the reference cell under more physiological conditions, in which inhibition strengthens as excitation increases. In earlier simulations, was manipulated by fixing the drive to inhibitory neurons (Eq 2). Here, the effective drive
was allowed to increase with
, mimicking increased drive to inhibitory neurons with rising stimulus intensity (Fig 3D, inset). The slope of this recruitment was controlled by varying
, which physiologically corresponds to changes in the efficacy of the afferent synapse onto inhibitory neurons. Increasing
shifted the
curve leftward (dotted to solid red, inset), causing the I–O curves of the reference neuron to become progressively more non-monotonic (Fig 3D). The curves shared a common threshold and overlapped at low
before diverging at higher values. Although these changes do not conform to classical forms of gain modulation, they can still generate multiplicative effects on tuned inputs (see below).
Correction for conductance effects.
A key assumption in Eqs 2 and 4 is the linear summation of excitatory and inhibitory inputs. This assumption is violated when synaptic inputs alter the total membrane conductance, since the net current at rest can differ substantially from that during depolarized states, leading to errors in predicted firing rates. To account for this effect, a conductance-dependent adjustment was derived (see S1 Appendix). Incorporating this correction substantially improved prediction accuracy (S2 Fig) while preserving the probabilistic formulation, allowing the same framework to be applied under conductance-based synapses.
Gain modulation of tuned responses.
The slope changes in the I–O curves described above (Fig 3B–3D) suggest mechanisms for multiplicative gain modulation of neural responses [27,28,52]. To test this, simulations were performed with following a Gaussian profile representing tuned sensory input (Fig 4A). Three conditions were examined: (mode 1) fixed
; (mode 2)
increasing linearly with
; and (mode 3)
following the I–O curve of the inhibitory neurons. In all cases, the peak input (
) produced modulated tuning curves with peak firing rates within 50% of the control (60 Hz), consistent with experimental data [27,28,52].
A, Simulations in which followed a Gaussian profile, representing a tuned sensory input. B, Case with constant
set to 0, 0.08, or 0.13. i, I–O curves used to transform the input. ii, Resulting tuned responses of the reference neuron. iii, Tuned responses normalized to a peak value of 1. C, Same as in B, except
increased linearly with
. This was implemented by setting
with
or 0.4 at a fixed
. Inset, Corresponding I–O curves across the full range of
. D, Same as in C, except
varied with
at different rates by setting
or 0.9.
For mode 1, relatively small values of were sufficient to produce multiplicative gain modulation. Although the I–O curves showed both slope and threshold changes (Fig 3B), the small range of
that was used primarily affected the slope (Fig 4B, left), resulting in a reduction in tuning curve amplitude (middle) with minimal change in width (right). Physiologically, such low
values (Eq 3) could arise from a small number of inhibitory neurons relative to afferents (low
), weak synaptic inhibition (low
), and/or low inhibitory firing rates (low
).
In mode 2, multiplicative gain modulation was achieved by setting the scalar in the relationship
to modest values. The resulting I–O curves were monotonic, with progressively reduced slopes as
increased (Fig 4C, left and inset). The peaks of the corresponding tuned responses (middle) varied with
and, when normalized, superimposed (right). To achieve a linear relationship between
and
under physiological conditions would require a fixed
across input intensities and low inhibitory firing thresholds to ensure engagement even for small
values.
In mode 3, multiplicative gain modulation occurred but only within a limited range of . At low
, the I–O curves overlapped substantially (Fig 4D, left), and divergence required strong inhibition at moderate
, where slope differences emerged. Under these conditions, the I–O curves became non-monotonic (inset). When
exceeded the range of the rising phase, firing shifted to the decaying portion of the curve, producing a central dip and bimodal tuning (S3 Fig). When restricted to the rising phase, however, peak responses could be modulated (middle) with minimal changes in tuning width (right).
Temporal firing profiles.
Neuronal firing patterns can encode distinct stimulus features or task-related components [36,53,54]. As the stimulus changes, the drive to the neuron—reflected in variations of —also changes. To examine how such temporal profiles depend on E–I interactions, it is necessary to systematically vary the time-dependent inputs. Controlling the inhibitory drive is challenging, however, because its onset and magnitude depend on how inhibitory neurons are recruited by the stimulus (Fig 2B). This recruitment, in turn, depends on the biophysical properties of inhibitory neurons [55] and on the amplitude of their excitatory inputs [56–58]. The firing profile of the reference cell also depends on the relative timing of EPSPs and IPSPs: in most cases, IPSPs lag EPSPs [2,18,40], although they can also precede them under certain conditions [59,60].
Replicating the full range of possible time-varying relationships between and
would require detailed modeling of inhibitory neuron dynamics, which is beyond the scope of this study. To allow direct and independent control of
, the inhibitory neurons were bypassed, and inhibitory inputs were generated in the same way as the excitatory afferents. The excitatory input, together with the conditioned inhibition
, was then delivered to the reference cell. This abstraction isolates the temporal interaction between excitation and inhibition, enabling their overlap to be examined in a controlled and systematic manner.
Both and
were ramped up to the same steady-state value and then ramped down (blue and red dashed curves in Fig 5A–5C, bottom panels). The simulation parameters were identical (see Fig 5 captions and Methods for details) except that the relative onsets were varied. The barrages were calculated as above and delivered to the LIF neuron, and firing histograms were compiled across repeated trials (top panels).
Ai, bottom: Excitatory and inhibitory input probabilities, (blue) and
(red), rose and fell together with no delay, each reaching a steady-state value of 0.5. The computed
is superimposed (black). Top: the corresponding spike histogram shows tonic firing in the reference neuron. ii–iii, Same as i, except
lagged or led
by 2 ms, respectively. B, Same as A, but with larger steady-state probabilities (0.75). C, Same as A, but with steady-state probabilities equal to 1. Model parameters: LIF neuron as in Methods. Per-afferent input intensities were
and
, with
Hz and population sizes
, bin width
ms.
A wide range of temporal firing patterns was generated by varying the magnitudes and relative timing of and
. In these simulations, the time courses of
and
were identical (Fig 5A–5C, bottom panels) but differed in their temporal lags (i–iii). For moderate inputs with no lag (A, i), the computed
(black) closely followed the time courses of
(blue) and
(red dashed), ramping to a steady level before decaying. The reference neuron responded with delayed tonic firing (top).
With larger input magnitudes (B), exhibited transient peaks at stimulus onset and offset, accompanied by a reduced steady-state level. Further increases in steady-state amplitude (C) eliminated the tonic component entirely, leaving only onset and offset peaks, since
only during those intervals. Accordingly, the reference neuron fired exclusively at stimulus onset and offset (top).
When lagged
by 2 ms (ii), similar results were obtained, except that the onset peak of
was larger than the offset peak (Fig 5C). This produced pronounced firing at stimulus onset—greater than that observed in the no-delay condition (i)—and no firing at stimulus offset. Conversely, when
preceded
(iii), spiking occurred only at stimulus offset.
Brief stimuli: Transient firing regime.
Brief stimuli, such as tone pips in the auditory system [2,33,59], light flashes in the visual system [61], or whisker deflections in the somatosensory system [4,6], evoke a volley of compound EPSPs, followed by compound IPSPs from inhibitory neurons after a small delay. Whether the postsynaptic cell fires depends both on the relative magnitude and timing of these inputs.
To examine the effects of E–I balance and timing on firing probability, simulations were performed in the same network as above. A brief stimulus evoked EPSPs in both the reference and I neurons whose arrival times followed a Gaussian distribution (Fig 6Ai, top panel, blue histogram). When the I neurons fired, they generated IPSPs in the reference cell a short time later, with a narrower temporal distribution (red). In response to these EPSPs and IPSPs, the reference cell fired action potentials with a narrower distribution (middle, gray) than the input, consistent with experimental observations [4].
Ai,top, probability distributions of arrival times of EPSP (blue) and IPSP (red) in the reference cell. Middle, predicted firing probability (magenta) overlaid on the spike time histogram (gray); Inset, magnified view. Bottom, superimposed traces of (blue),
(red), net input
(black), and threshold-crossing probability (green).
,
. ii, same but with
. B,i, I-O curves showing mean firing probability (±SD) vs.
, for fixed
values (
). The level was specified by adjusting and then fixing
. Other parameters are kept constant:
pA, which produced PSPs with amplitude
. Predicted values (magenta) are overlaid. Increasing
shifts the curve rightward without changing slope. Inset, I-O curves aligned by threshold. ii, same but with
increasing linearly with
, implemented by setting
. iii, same but with
. Model parameters:
,
. Histograms compiled over 5000 trials with Bin width =
ms.
The instantaneous excitatory and inhibitory probabilities (blue and red traces in Fig 6A, bottom panel) were computed from the compound EPSPs and IPSPs using convolution expressions derived in S1 Appendix. These are given by the probability densities
where is the excitatory arrival-time density and
is the conditioned inhibitory arrival-time density, both obtained by convolving peak-normalized unitary PSPs with the corresponding spike-time histograms. The effective probability
is as defined above, except that the ratio of inhibitory to excitatory amplitudes
replaces the ratio of charges
. The time-varying excitatory (blue) and inhibitory (red) probabilities thus follow the same temporal profiles as the compound EPSP and IPSP, respectively (Fig 6Ai, bottom panel).
The instantaneous probability density of net excitation—that is, the component of the excitatory drive that is not canceled by inhibition (black curve in Fig 6A, bottom panel)—is defined as
When excitation is present, this expression can be equivalently rewritten as
where the ratio quantifies the instantaneous balance between inhibition and excitation. This time-dependent formulation is directly analogous to the sustained-stimulus case, where
represents the steady-state probability that excitation survives inhibition.
The difference form provides the primary definition of and remains well defined throughout the stimulus, including times at which excitation vanishes. To preserve its interpretation as a probability density, negative values are rectified to zero, ensuring
for all t.
The timing of action potentials is computed in two stages (see S1 Appendix). First, the probability that the net excitatory input at a given time exceeds the threshold required for firing was computed from (bottom, green curve; S29 Eq of S1 Appendix). Second, this probability was used to derive the expected firing time (S30 Eq of S1 Appendix), accounting for the constraint that a neuron can fire only once and only if it has not already fired. This procedure yields the predicted firing probability (middle panel, magenta curve), which closely matches the spike histogram (gray, inset). Similar results were obtained for a weaker stimulus (Fig 6Aii) and when the synaptic inputs were delivered as conductances (S4 Fig).
Modulating inhibitory strength led to changes in the I–O curves that deviated from model predictions. The total probability of firing was calculated either from the area under the spiking histogram (gray) or from the predicted probability (magenta). Fig 6Bi shows
plotted against
for different levels of
, analogous to the green curves in Fig 1B. The value of
was controlled by adjusting and then fixing
in Eq 3 so that
remained constant across values of
. As
increased, the level of
required to evoke firing also increased, resulting in a rightward shift of the I–O curves without a change in slope. When aligned by their respective thresholds, the curves collapsed onto a single profile (inset). Thus, unlike in the toy model or under sustained input conditions, the effect on the I–O relationship was purely additive.
Similar results were obtained when increased linearly with (as in Fig 3C). This was implemented by letting increase as
. As the slope of vs. increased, the curves shifted to the right (Fig 6Bii), but unlike the toy model (Fig 1B, orange curves) and tonic input (Fig 3C), remained monotonic with no change in slope (inset). Thus, the effect on the I–O curve is also additive.
In Fig 6Biii, the inhibitory input probability was modulated by varying
so that, as in the sustained-stimulus condition (Fig 3D, inset), the resulting
reflected the I–O function of the inhibitory (I) neurons. As
increased, the slopes of the I–O curves of the reference cell decreased, whereas their thresholds remained constant. Unlike in the simple model, the curves remained monotonically increasing, with those obtained under weaker inhibition saturating at high
. Thus, under these conditions, the effect on the I–O curve was predominantly multiplicative within the range preceding saturation (inset).
Taken together, these results show that for transient inputs the effect of inhibition on firing probability depends on how inhibitory recruitment interacts with threshold crossing during the transient (see S1 Appendix for a detailed analysis). When inhibition was independent of excitation or scaled proportionally with it (Fig 6Bi–ii), increasing inhibitory strength primarily shifted the effective threshold without altering the slope of the input–output relation, whereas feedforward recruitment of inhibition (Fig 6Biii) also changed the sensitivity of firing probability to , producing multiplicative gain modulation.
Discussion
This study aimed to elucidate the principles governing stimulus-driven excitatory–inhibitory (E–I) interactions in feedforward inhibitory circuits. By combining excitatory and inhibitory effects within a probabilistic framework, a simple relationship was derived that predicts how E–I balance shifts during stimulation. This formulation links those shifts to both cellular and network-level variables, offering mechanistic insight into how inhibition shapes neuronal I–O functions, modulates gain, and influences overall response dynamics.
A probabilistic formulation offers several theoretical advantages over classical approaches that rely on summing synaptic currents to predict firing [51]. First, the multiplicative expression introduced here captures the survival of excitatory inputs under feedforward inhibition in a form that is both biologically interpretable and analytically tractable. Because probabilities are inherently bounded between 0 and 1, the resulting I–O curves are automatically rectified and sigmoidal—features that are often imposed manually in models to prevent negative or unbounded firing probabilities or rates [10,19,30,62–64].
Second, the net probability can be used directly in analytic expressions to derive I–O relationships and characterize gain control. The reduced form naturally incorporates the variables that determine
(such as interneuron recruitment and synaptic reliability) and automatically scales with
, eliminating the need for manual adjustment when analyzing I–O curves. Because
closely tracks temporal firing patterns, it can be used directly, or as input to Poisson point-process models [65], to efficiently predict stimulus-evoked responses with realistic temporal statistics.
Third, this framework explicitly incorporates synaptic noise, yielding smoothly rising I–O curves without additional assumptions. The sources of variability are well defined and can, in principle, be used to compute statistical properties of both inputs and outputs, linking fluctuations in synaptic drive to the variability and reliability observed in neuronal firing.
Relation to previous work
The model’s I–O curves exhibit two distinct regimes. For weak stimuli (low ), firing occurs in a sub-oscillatory regime driven primarily by input variability, producing a sigmoidal rise. This behavior aligns with observations in visual cortex, where input summation is supralinear for weak stimuli and sublinear for strong ones [64,66]. The underlying mechanisms may differ, however, since the cortical responses reflect recurrent connectivity, which is absent in the present feedforward model. As input strength increases, the system transitions to an oscillatory regime, in which the I–O curve may continue to rise, flatten, or become non-monotonic. The specific shape of the curve is determined by how inhibition scales with excitation, consistent with previous modeling results [19].
The model highlights key variables that govern the transformation of synaptic input into spiking output. Understanding this transformation has practical implications, as it may enable the inference of intracellular dynamics from extracellular recordings [67]. Although direct inversion of the model equations is not feasible due to the number of interacting variables (e.g., ,
,
,
,
), the derived relationships can nevertheless inform which parameters must be independently measured or constrained to obtain reliable estimates.
Gain modulation
Multiplicative gain modulation of I–O curves is essential for maintaining tuning acuity across different cognitive or behavioral states [27,28,68]. The present model reveals multiple mechanisms for achieving such modulation by varying how scales with
(Fig 3B–3D). Unlike previous models [10,13,30], these effects arise without invoking recurrent connectivity, synaptic noise, or conductance-based mechanisms. Moreover, the framework provides a means to predict neuromodulatory influences based on their actions on synaptic or biophysical parameters [69,70].
For brief stimuli, in which firing is strongly influenced by the delay between excitation and inhibition [4,7,59], multiplicative gain emerged only when the drive to inhibitory neurons increased with (Fig 6Biii)—a condition likely to occur with natural stimuli. Optogenetic activation of inhibitory neurons during brief sensory stimulation [33] most closely resembles the simulations with fixed
(Fig 6Bi), which predict additive modulation of the I–O curves. The mixed multiplicative and additive effects observed experimentally may therefore arise because optogenetic stimulation recruits inhibitory neurons in a manner that is partially decoupled from their natural, stimulus-dependent activation, placing the system in an intermediate regime between the scenarios considered here. For prolonged stimuli, the model predicts that modulation is predominantly multiplicative, consistent with observations during sustained optogenetic activation of inhibitory neurons [34].
More broadly, these results highlight a fundamental distinction between gain control under transient and sustained stimulation. For sustained inputs, excitation and inhibition reach a steady-state balance, so changes in inhibitory strength directly rescale the effective drive and naturally give rise to divisive gain modulation. For transient inputs, however, firing is dominated by threshold crossing within a brief temporal window, and inhibition can influence firing either by delaying this crossing—producing additive shifts of the I–O relation—or, when inhibitory recruitment itself scales with excitatory drive, by altering the sensitivity of firing probability to further increases in , resulting in true multiplicative gain modulation.
Temporal response patterns and non-monotonic input–output relations
The model reproduced many of the temporal firing patterns observed in cortical and subcortical regions [35–39,46,53] by adjusting the magnitude and timing of the excitatory and inhibitory probabilities (Fig 5). A key prediction is that transient firing and non-monotonic I–O curves observed experimentally [16,35] need not arise from disproportionately strong inhibition. Both phenomena can emerge even when and
increase in a balanced or proportional manner (Fig 2B, Fig 3C–3D, Fig 5).
The model further predicts that onset–offset responses can occur when both excitatory and inhibitory probabilities approach unity, with small shifts in inhibitory delay producing distinct temporal response components (Fig 5; [42]). Whether similar mechanisms operate in cortex remains unclear, as onset and offset responses may be inherited from subcortical pathways [40,53] or generated de novo within cortical circuits [22,41,44].
Limitations
The theory relies on several key assumptions. First, the calculation of assumes that inhibitory inputs sum linearly with, and effectively cancel, excitatory inputs. Although the sublinear summation associated with conductance changes can be minimized through a correction term, the model does not account for heterogeneity in the amplitudes of individual EPSPs and IPSPs. For analytical simplicity, IPSP amplitudes were expressed as fixed fractions of the corresponding EPSPs. In reality, postsynaptic potential size depends on several factors, including synaptic location and the identity of the presynaptic population. For example, thalamocortical synapses are typically stronger than cortico-cortical synapses [71]. Moreover, IPSPs at the soma can, in principle, shunt or cancel multiple small EPSPs originating in distal dendrites.
Second, the constants () that scale
were chosen so that their product lies between 0 and 1, ensuring that
remains interpretable as a probability. Although precise values for these constants are unlikely to be fully known in any given system, reasonable estimates can be drawn from the literature (
, [72,73];
, [8,9,74];
, [8,9,74];
, [49,57,75]). In addition,
or
may vary dynamically, reflecting short-term plasticity of excitatory and inhibitory synapses [9,55]. For modeling purposes, parameters should be constrained so that the product remains below unity. As an additional safeguard, a smooth saturating transform, such as
, can be applied to preserve the probabilistic interpretation of
even when the product exceeds one.
Third, the model assumes that recurrent connections do not substantially contribute to the evoked firing rate. This assumption is likely valid for brief stimuli, during which recurrent activity has limited influence on firing [76]. In the cortex, the connection probability between excitatory neurons is relatively low [9,55]. Consequently, if the active region is spatially restricted—as in tuned responses driven by topographically organized inputs—the small number of recruited local excitatory neurons would exert minimal impact, consistent with simulation results [76]. Nonetheless, incorporating recurrent connectivity will be important for extending the model to more general network configurations.
Fourth, spontaneous activity—whether intrinsically generated or originating from neurons outside the feedforward pathway—is not explicitly included in the present analysis. Background activity can alter the initial conditions of the network, influence transient responses [77,78], and contribute to steady-state firing. In principle, such effects could be incorporated into the probabilistic framework if expressed in terms of effective input probabilities, although this would require additional theoretical development.
Finally, a potential limitation of the framework is that spiking is described as being determined solely by inputs within the current time bin, without explicit dependence on prior activity. This simplification is partly mitigated by the model’s structure. For brief stimuli, the probability of a spike at a given time is defined conditionally on no spikes having occurred in earlier bins, thereby incorporating past spiking history implicitly into the first-spike probability. In addition, the time-dependent changes in probability follow the shape of the underlying synaptic potentials, which reflect recent input dynamics. For long-duration stimuli, the analysis focuses on the average synaptic current, which depends primarily on the overall rate and distribution of synaptic events. In this regime, the precise temporal sequence of individual inputs becomes less critical, as the mean current is determined by their statistical properties.
Despite these limitations, the model effectively captures many key aspects of stimulus-evoked responses and can serve as a foundation for developing formal mathematical analyses [79] to study more complex network configurations.
Materials and methods
Network parameters
The circuit consists of a reference neuron, whose firing activity serves as the model’s output, and 20–50 local inhibitory neurons. Both cell types received afferents from an excitatory source, with the reference cell also receiving inputs from
interneurons. Typical values of the variables used in the simulations are listed in Table 2.
Leaky integrate-and-fire parameters
Simulations were performed in the Matlab programming environment. Neurons were modeled as standard leaky integrate-and-fire (LIF) units governed by
where MΩ,
mV is the resting potential,
ms is the membrane time constant,
pF is the capacitance, and the action potential threshold was set to
mV. After an action potential, the membrane potential was reset to
. The terms
and
denote the total excitatory and inhibitory synaptic currents, respectively (in nA). The integration time step (bin width) was
ms.
Each unitary synaptic current , evoked by a single presynaptic spike, was described by a scaled alpha function:
where ms and the peak amplitude was normalized to 1. For current-based simulations,
pA and
pA. For conductance-based simulations,
nS and
nS, with reversal potentials
mV and
mV. Under these conditions, the corresponding excitatory and inhibitory postsynaptic potentials (PSPs) had amplitudes of approximately
V and
V, respectively, when measured at the resting potential.
Stimulus parameters
Sustained stimulus.
For sustained stimulation, excitatory synaptic barrages to the reference and inhibitory neurons were generated by creating a spike train according to a Poisson process with total rate , where
is the average firing rate of a single afferent and
is the number of afferents. The excitatory barrage to the inhibitory cells was adjusted by scaling (multiplying) the total rate by
. The excitatory barrage (0.1–1 s duration) was then obtained by convolving the spike trains with
.
To construct the inhibitory barrage, the excitatory barrages were delivered to (
) inhibitory cells, and the resulting spike trains were stored in a matrix. From this set,
trains were randomly selected, summed, and subsequently convolved with
. This inhibitory barrage was then delivered together with the excitatory barrage to the reference neuron. The average firing rate and peristimulus time histograms of the reference neuron were computed over 100–10000 trials, with each trial using different realizations of the synaptic barrages.
To examine gain modulation of I-O functions (Fig 4), the excitatory input probability was modeled as a Gaussian function,
where denotes the effective excitation probability at stimulus feature x,
is the preferred stimulus, and
is the tuning width. This spatially tuned
was then used to generate the excitatory drive to both the reference cell and the inhibitory neurons, following the procedures described above. Average excitatory and inhibitory synaptic currents, together with the evoked firing rate, were computed over 100 trials and plotted as functions of x.
In a subset of simulations (Fig 5), the inhibitory population was bypassed. In these cases, the inhibitory input was generated using the same procedure as for the excitatory input and, after appropriate scaling, was delivered directly to the reference neuron. Both excitatory and inhibitory spike trains were modeled as inhomogeneous Poisson processes with time-varying probabilities, producing instantaneous rates
The functions and
shared the same temporal profile except for a fixed relative delay. Both ramped linearly from 0 to 1 over a 20 ms period, maintained the steady-state value for the duration of the stimulus, and then decayed back to zero. The inhibitory delay, defined as the onset difference between
and
, was varied between
and
ms. Firing rates were set to
, and the numbers of inputs were
. Peristimulus time histograms (PSTHs) of the reference neuron were computed from 100–1000 independent trials.
Transient stimuli.
In this mode, each afferent fired a single action potential in response to a brief stimulus. Temporal jitter was introduced so that the afferent spikes did not arrive synchronously across inputs. The distribution of excitatory postsynaptic current (EPSC) arrival times to the reference and inhibitory neurons, denoted , was modeled as a Normal distribution scaled by
:
Here, and
represent the effective distributions of EPSC arrival times in the reference and inhibitory neurons, respectively, accounting for the fact that not all afferent spikes evoke synaptic currents unless
or
equal to 1.
Each histogram was convolved with the unitary synaptic kernel , which defines the time course of a single postsynaptic current, to obtain the compound excitatory current in the reference and inhibitory neurons. Independent realizations of
were generated for each inhibitory cell, and their evoked spike times were documented.
A corresponding histogram of inhibitory spike times, , was compiled from the responses of the inhibitory neurons. This histogram was scaled by
and convolved with
, which defines the time course of a unitary inhibitory synaptic current. Because inhibitory activity was estimated conditionally on stimulus-evoked inhibition (see S1 Appendix), the resulting current was delivered directly to the reference neuron together with the excitatory current. Each simulation was repeated 1000–5000 times to compile the peristimulus spike histogram of the reference neuron (Fig 6A, middle panel, gray).
Inhibitory timing was estimated from trials in which inhibitory spiking occurred (i.e., conditioning on stimulus-evoked inhibition), yielding and
; consequently, no additional factor of
is applied.
Methods for calculating the time-dependent probabilities and predicted spike times are provided in S1 Appendix.
Supporting information
S4 Fig. Transient response in conductance mode.
https://doi.org/10.1371/journal.pcbi.1013958.s004
(TIF)
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