One-compartment neuron model

The dynamics of membrane potential, V, in the one-compartment neuron model are (1) where the membrane capacitance is C = 1 μF/cm². Here, and throughout, we use V′ to indicate the time-derivative of V, i.e. dV/dt. The ionic currents (leak, potassium, A-type potassium, and sodium) are given by the equations (2) We use the following fixed parameter values for maximal conductances: g_L = 1 mS/cm², g_K = 45 mS/cm², and g_Na = 37 mS/cm². We use a range of values for the maximal conductance of the A-current (g_A) to observe transitions between subtractive and divisive effects of inhibition. The reversal potentials are V_L = −70 mV, V_K = −80 mV, and V_Na = 55 mV.

We make several simplifications, similar to those first suggested in [22], to the gating variables in the model. We identify sodium activation as a fast process and assume it evolves instantaneously to its voltage-dependent steady-state value. That is, we let m = m_∞(V) = 1/(1 + e^−(V+30)/15). In addition, we observe an approximately linear relationship between sodium inactivation and potassium inactivation and thus set h = 1 − n.

The remaining gating variables are n, a, and b. Their dynamics are described by equations of the form (3) The voltage-dependent steady-state functions are of the form . For the potassium activation variable, n, we assume that ϕ_n = 0.75, θ_n = −32 and σ_n = −8. The time-scale for the n variable is voltage-dependent: τ_n(V) = 1 + 100/(1 + e^(V+80)/26). Similar to the model presented in [23], we assume that ϕ_a = 1, θ_a = −50 and σ_a = 20 for A-type potassium activation, and ϕ_b = 1, θ_b = −70 and σ_b = −6 for A-type potassium inactivation. The time-scales for the A-type current are constants: τ_a = 2 ms and τ_b = 150 ms.

Inputs to the model include synaptic excitation (I_Syn,E) and inhibition (I_Syn,I). Excitatory current is I_Syn,E = g_Syn,E s_E(V − V_E) and inhibitory current is given by an analogous equation. The maximal excitatory and inhibitory conductances (g_Syn,E and g_Syn,I) are parameters that we vary in simulations. The reversal potentials are V_E = 0 mV for excitation and V_I = −85 mV for inhibition. The gating variables, s_E and s_I, reset to one at the time of a synaptic event and decay with an exponential time-course. That is, the excitatory gating variable is defined as (4) where t_E is the time of the most recent excitatory event and the decay time constant is β_E = 0.2 ms⁻¹. A similar equation holds for the inhibitory gating variable s_I, but with a time constant β_I = 0.18 ms⁻¹. Excitatory event times are randomly distributed according to a homogeneous Poisson process with rate r_E. Inhibitory event times are periodic with rate r_I. We vary the values of the rate parameters (r_E, r_I) in our investigations. Our choice of these input patterns simplifies some of our mathematical analysis. In addition, our choice of periodic inhibitory events was motivated by the design of in vitro experiments, presented in [24], in which inhibitory interneurons were activated periodically using optogenetic techniques. In additional simulations included as S1 and S2 Figs we allowed the timing of inhibitory inputs to be random with event times drawn from a homogeneous Poisson process. We did not observe substantially different results in simulations that used these non-periodic inhibitory inputs.

Multi-compartment neuron model

In some simulations we augment the one-compartment (point neuron) model by attaching additional compartments that represent a dendritic process. We assume that the dendrite consists of nine equally-sized compartments. Moreover, the neuron receives inhibitory input at its soma (the first compartment) and excitatory input at a dendritic compartment.

Voltage in the first compartment (soma) is denoted V₁ and is given by Eq 1 with synaptic excitation removed and with a new term representing the flow of current between compartments (axial current): (5) The remaining dendritic compartments do not include potassium, A-type potassium, or sodium currents, and thus V_j for 2 ≤ j ≤ 10 follows the linear dynamics of a passive cable: (6) Leak conductance in the dendrite compartments is g_L = 0.1 mS/cm² (one-tenth the value in the first compartment). Axial current is (7) where g_Ax = 10 mS/cm².

Input currents are defined in a manner identical to inputs in the one-compartment model. Excitatory and inhibitory gating variables follow Eq 4. Excitatory synaptic event times are drawn from a homogeneous Poisson process with rate r_E and inhibitory synaptic event times are periodic with rate r_I. These constants, as well as synaptic input strengths (g_Syn,E, g_Syn,I) and the compartment targeted by the excitatory inputs, are parameters we vary in our investigations.

Computations

We simulated the point-neuron and multi-compartment neuron models using software written in the C computer programming language. We integrated differential equations using the fourth order implicit Runge-Kutta method available in the GNU scientific library. We also simulated the one-compartment model and a reduced model version of the one-compartment using XPPAUT, and performed bifurcation analysis of these models using the AUTO feature of XPPAUT [25]. Simulation code is available for download at https://github.com/jhgoldwyn/Gain-Control-With-IA.

Examples of divisive and subtractive inhibition

We first study the relationship between excitatory input rate (r_E) and firing output rate (r_out) of the one-compartment model. In Fig 2A, we plot examples of this input/output relationship for simulations without inhibition (empty circles, g_Syn,I = 0) and with inhibition (filled circles, g_Syn,I = 1). The A-channel conductance in these simulations is g_A = 20 mS/cm². For these parameters, we observe that inhibition reduces the model neuron’s output firing rate, but the neuron continues to fire in response to arbitrarily low input rates.

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Fig 2. Examples of divisive and subtractive effects of inhibition in the one-compartment model.

A, B: Output firing rates as a function of excitatory input rate, computed from simulations without inhibition (empty circles, g_Syn,I = 0) and with inhibition (filled circles, g_Syn,I = 1 and r_I = 50 Hz). Excitatory synaptic strength is g_Syn,E = 0.5. In A: Divisive rescaling of the input/output relation with g_A = 20. In B: Subtractive shifting of the input/output relation with g_A = 40. C: Data from A and B are replotted with output firing rates in the absence of inhibition on the ordinate and output firing rates in the presence of inhibition on the abscissa. Threshold-linear functions are fit to simulation data (black lines). Rightward shift of threshold-linear function for g_A = 40 is characteristic of subtractive inhibition.

https://doi.org/10.1371/journal.pcbi.1006292.g002

An additional way to view the effect of inhibition is to plot output firing rates in the presence of inhibition as a function of output firing rates in the absence of inhibition, as we have done in Fig 2C. There is a roughly linear relationship between these output firing rates, which we describe by fitting these data with a threshold-linear function of the form (8) where the symbol [⋅]₊ indicates we set y = 0 if the argument m(x − x₀) is negative. We obtain the slope parameter m and the x-intercept parameter x₀ by applying a curve-fitting procedure (using the fminsearch command in MATLAB) to the portion of data for which the output firing rate in the presence of inhibition is less than five spikes per second. In this example, inhibition affects the value of the slope parameter m, but the value of x₀ is nearly zero. We identify responses with these characteristics as cases in which the effect of inhibition is divisive.

In Fig 2B, we increase the A-channel conductance to g_A = 40 mS/cm². We observe that inhibition has a different effect on the input/output curve in these simulations. In the presence of inhibition (filled circles), there is now a non-zero value of the input rate below which the neuron model does not spike (r_out = 0 for r_E ⪅ 30). Moreover, when we view the relationship between output firing rates with and without inhibition in Fig 2C, we observe a rightward shift of the threshold-linear function fit to these data (positive-valued x-intercept). We identify responses with these characteristics as cases in which the effect of inhibition is subtractive.

Although we refer throughout to the effect of inhibition on responses as being either divisive or susbtractive, this is a simplification of a more complicated and subtle reality. In fact, responses can show characteristics of both divisive and subtractive inhibition. In particular, the input/output curves can be right-shifted (evidence of subtractive inhibition) and have slopes that are decreased relative to slopes for g_A = 0 (evidence of divisive inhibition). We provide evidence of such “mixed” responses in S3 Fig. To be clear: we refer to scenarios in which the input/output curves have only a change of slope as divisive, and scenarios in which the input/output curve is right-shifted as subtractive. In other words, the subtractive case will also include “mixed” responses.

Parameter study: Inhibition is subtractive for strong A-current conductance or weak excitatory conductance

We identify two parameters in the one-compartment model that are key factors in determining whether inhibition has a divisive or subtractive effect on firing rate responses: the A-channel conductance (g_A) and the excitatory synaptic conductance (g_Syn,E). In Fig 3A we show a set of threshold-linear functions computed using g_A = 20, 30 and 40, and synaptic excitation strength fixed at g_Syn,E = 0.5. The transition from divisive to subtractive inhibition is evident in the rightward shift of these threshold-linear functions with increasing values of g_A. This transition occurs, for this parameter set, for g_A ≈ 33, a point we investigate in more detail below, with simulations and phase plane analysis.

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Fig 3. Inhibition is subtractive for large A-channel conductance or weak synaptic excitation.

A, B: Firing rates computed from simulations with inhibition (g_Syn,I = 1, r_I = 50 Hz, abscissa) plotted as a function of firing rates computed from simulations without inhibition (g_Syn,I = 0, ordinate). In A: Three values of A-channel conductance are compared (g_A = 20, 30, 40) with synaptic excitation strength fixed at g_Syn,E = 0.5. Inhibition is subtractive for large g_A evident in the rightward shift of the threshold-linear relationship between firing rates for g_A = 40. In B: Three values of synaptic excitation strength are compared (g_Syn,E = 0.4, 0.5, 0.7) with A-channel conductance fixed at g_A = 30. Inhibition is subtractive for weaker excitation, evident in the rightward shift of the threshold-linear relationship between firing rates for g_Syn,E = 0.4.

https://doi.org/10.1371/journal.pcbi.1006292.g003

In Fig 3B, we show a set of threshold-linear functions with g_A = 30 fixed, but now varying the value of g_Syn,E from 0.4 to 0.7. The stronger excitatory inputs (g_Syn,E = 0.5, 0.7) cause inhibition to have a divisive effect, while the weaker excitatory input (g_Syn,E = 0.4) causes inhibition to have a subtractive effect. In these simulations, we do not vary the parameters associated with inhibition. They are g_Syn,I = 1 and r_I = 50 Hz.

From these simulations, we conclude that the effect of inhibition on firing rates in the one-compartment model can switch from divisive to subtractive for sufficiently strong A-current conductance or sufficiently weak excitatory synaptic conductance. In the parameter plane of g_A and g_Syn,E, then, there is a boundary that separates parameter sets that produce divisive inhibition from parameter sets that produce subtractive inhibition. We map this boundary by performing simulations throughout the (g_A, g_Syn,E) parameter space. For each simulation, we fit threshold-linear functions to characterize the relationship between output firing rates in the presence and absence of inhibition. We then find the smallest value of g_A for which the x-intercept of the threshold-linear function is right-shifted by more than two spikes per second and label this as boundary between subtractive and divisive inhibition.

In Fig 4, we show the results of this parameter exploration. We performed these simulations and classification procedure for several values of inhibition conductance strength (varying values of g_Syn,I, in Fig 4A), and for several values of inhibition rate (varying values of r_I, in Fig 4B). The lines in each panel separate parameter regions for which inhibition is divisive (lower right corners in each panel) from parameter regions in which inhibition is subtractive. This confirms our earlier observation that the effect of inhibition is subtractive if A-channel conductance is sufficiently strong or excitatory inputs are sufficiently weak.

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Fig 4. Boundary between subtractive and divisive inhibition in (g_Syn,E, g_A) parameter space.

A, B: For each parameter set, we fit threshold-linear functions to characterize the relationship between output firing rates in the presence and absence of inhibition. Dots in each panel identify the smallest value of g_A (for a given parameter set) at which inhibition is subtractive. In A: We vary inhibition strength (g_Syn,I = 0.5, 1, 2) and keep inhibition rate fixed at 50 Hz. In B: We vary inhibition rate (r_I = 30, 50, 70 Hz) and keep inhibition strength fixed at g_Syn,I = 1. The values of g_A that define the boundary between subtractive and divisive inhibition decrease with increases in either inhibition parameter (g_Syn,I or r_I).

https://doi.org/10.1371/journal.pcbi.1006292.g004

These simulations also demonstrate that inhibition parameters modify (weakly) the location of the boundary between divisive and subtractive inhibition in the (g_A, g_Syn,E) parameter plane. Stronger inhibition (either through larger g_Syn,I or larger r_I values) decreases the portion of the (g_A, g_Syn,E) parameter plane in which inhibition has a divisive effect on firing rate responses.

Analysis of a reduced one-compartment model

We use mathematical analysis to derive the parameter regions in which the model exhibits either a divisive or subtractive response to inhibition. We begin by considering a reduced model in which activation of the A-current is instantaneous; that is, a = a_∞(V). Later, we discuss how the model’s response to inhibition may change if this assumption does not hold.

Excitability analysis using fast/slow dissection.

A first step in the analysis is to determine under what conditions the neuron will fire an action potential in response to an excitatory input. An important distinction between divisive and subtractive inhibition is whether the neuron can respond to arbitrarily low excitatory input rates. We analyze the model by viewing solutions in the (V, n) phase plane. A major challenge in this approach is that the phase plane depends not only on the other dependent variable, b, but on the values of the synaptic inputs, s_E and s_I. Suppose, for the time being, that these variables are fixed constants. Fig 5 shows phase planes for different values of these constants. In particular, it illustrates how the V-nullclines change as b, s_E, s_I and the parameter g_A changes. In each case, the V-nullcline has left, middle and right branches, while the n-nullcline is a monotone increasing function. Note that values of n along the V-nullcline are increasing functions of s_E and decreasing functions of b, s_I and g_A.

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Fig 5. Dependence of the V-nullcline on A: g_A, B: b, C: s_I and D: s_E.

Default values of the parameters are g_A = 20, b = .5, s_I = .5 and s_E = 1. Moreover, g_Syn,E = 3 and g_Syn,I = 5. Thin blue line is n_∞(V), the n-nullcline.

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We consider V to be a fast variable and n and b to be slow variables. During silent or active phases, solutions lie on, respectively, the left or right branch of the V-nullcline corresponding to values of b, s_E and s_I. The jumps up and down of action potentials correspond to horizontal transitions between left and right branches in the phase plane.

Now suppose that the solution initially lies in the silent phase along the left branch of some V-nullcline and there is an excitatory synaptic input. Then s_E will immediately jump from s_E = 0 to s_E = 1, resulting in an immediate change in the V-nullcline, as shown in Fig 5D. If (V(0), n(0)) lies below the left knee of this new s_E = 1, V-nullcline, then the solution will jump to the right branch of the new V-nullcline, resulting in an action potential. However, if (V(0), n(0)) lies above this left knee, then the solution will jump to the left branch of the new V-nullcline; that is, the solution will not respond to the excitatory input with an action potential.

This discussion helps to explain when the neuron will or will not fire an action potential in response to an excitatory input. In particular, there are two reasons why the neuron may not respond and these are illustrated in Fig 6. The first reason is that the neuron may be in a refractory period. That is, if the excitatory input arrives shortly after the neuron has previously fired, then the K⁺ activation variable, n, may not have had enough time to recover and evolve below the left knee of the s_E = 1 cubic nullcline. The second reason is that the inhibitory input, s_I, may be too strong and the left knee of the s_E = 1 cubic may lie below the n = 0 axis. In this case, the neuron will not respond even though there has been a long time since the preceding excitatory input.

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Fig 6. Response to an excitatory input.

A. The neuron will or will not fire an action potential if, at the time of the excitatory input, it lies below or above the left knee of the s_E = 1 cubic, respectively. B. The neuron cannot respond with an action potential if the left knee of the s_E = 1 cubic lies below the n = 0 axis.

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Estimate of b. One difficulty is that the cubic nullclines depend on the time-dependent variable b. For the analysis, we replace b by its average value along solutions, which we estimate as follows. A key observation is that the average value of b can be well approximated by considering a model consisting only of synaptic inputs and the leak current. In particular, to compute the average value of b we may ignore the voltage dependent Na⁺, K⁺ and A-currents.

To justify this claim, we first simulate the full model for different values of g_A and r_E, and then compute the average value of b along solutions, which we denote as b_av(r_E, g_A). Fig 7A shows plots of b_av versus r_E for different values of g_A. Note that b_av depends only weakly on g_A and is a decreasing function of r_E.

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Fig 7. Approximation of b (a slow variable) by its average value.

A: Plots of b_av vs. r_E for different values of g_A. B: Plots of , and b_av(r_E, g_A) with g_A = 40. In both panels, g_Syn,E = 3, g_Syn,I = 5 and r_I = 50 Hz.

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We next repeat these simulations with g_A = g_K = g_Na = 0. Actually, we perform the simulations twice: once with purely excitatory inputs (r_I = 0) and then with purely inhibitory inputs (r_E = 0). This gives rise to a curve and a constant . We then let Fig 7B shows that gives an excellent approximation of b_av(r_E, g_A).

We can simplify the anaysis further by considering a fast/slow reduction with V as the fast variable. If g_A = g_K = g_Na = 0, then V satisfies the linear equation: Treating V as a fast variable, we set the right hand side of this equation equal to 0 and solve for V to obtain (9) As before, we treat excitatory inputs and inhibitory inputs separately. Let Then let and be solutions to Eq 3 with X = b, and V = V_E and V = V_I, respectively. Finally, let and be the average values of and along solutions and set Fig 7B shows that gives an excellent approximation of . In the analysis that follows, we replace the dependent variable b by constants .

The left knee. Our previous discussion emphasized the importance of the left knees of the cubic-shaped V-nullclines when s_E = 1. If this left knee lies below the n = 0 axis, then the neuron cannot spike in response to an excitatory input. For the reduced model, with a = a_∞(V), this left knee depends on the values of s_I, b and g_A. Here, we replace b by and denote the value of n at the left knee as N_lk(s_I, r_E, g_A). We compute the positions of the left knees using XPPAUT.

Identification of inhibition as divisive or subtractive. By computing the position of the left knee, we can determine for which parameter values inhibition will have a divisive effect and for which values it will have a subtractive effect. Let P_I be the period of inhibitory inputs. Since between inhibitory inputs, it follows that (10) for all t.

Fig 8A shows plots of N_lk versus g_A when r_E = 0 and s_I = σ_*. Note that N_lk is a decreasing function of g_A and there exists such that . Since N_lk is an increasing function of r_E, it follows that if , then N_lk(σ_*, r_E, g_A) > 0 for all r_E > 0. Since N_lk is a continuous function of s_I, it follows that if and s_I > σ_* with s_I − σ_* sufficiently small, then N_lk(s_I, r_E, g_A)> 0 for all r_E. In this case, the neuron is able to respond to arbitrarily low firing rates and we expect the response to inhibition to be divisive.

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Fig 8. Identifying g_A value at the point of transition between divisive and subtractive inhibition.

Dependence of the left knee, N_lk(s_I, r_E, g_A), on A: g_A with s_I = σ_* and r_E = 0; and B: r_E with s_I = σ_* and different values of g_A. Here, . C: Theoretical calculation of Γ(g_A) (solid lines) and measured values of the minimum input rate at which solutions of the reduced model exhibits non-zero output firing rates (small triangles) for different values of the inhibitory input rate, r_I.

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Fig 8B shows plots of N_lk versus r_E for different values of g_A when s_I = σ_* Note that if , then there exists a critical value of r_E, which we denote by Γ(g_A), such that if r_E < Γ(g_A), then N_lk(σ_*, r_E, g_A)<0; if r_E > Γ(g_A), then N_lk(σ_*, r_E, g_A)>0. Recall that N_lk(s_I, r_E, g_A) is a decreasing function of s_I. Hence, if and r_E < Γ(g_A), then N_rk(s_I, r_E, g_A) < 0 for all s_I > σ_*. Moreover, if and r_E > Γ(g_A) then N_rk(s_I, r_E, g_A) > 0 for s_I > σ_* with s_I − σ_* sufficiently small. Hence, if , the neuron is not able to respond to arbitrarily low firing rates and we expect the response to inhibition to be subtractive. Given , the minimum input rate at which the neuron can respond is Γ(g_A). Fig 8C compares our theoretical calculation of Γ(g_A) to measured values of the minimum input rate at which the reduced model exhibits non-zero output firing rates (theoretical approximation shown with solid line, simulation results shown with symbols). We report this comparison between theory and computation for values of inhibitory input rate (r_I) that vary from 30 Hz to 70 Hz in increments of 10 Hz.

Output rate approximation using dead-time modified Poisson process.

The previous analysis was concerned with small output firing rates. Here we consider output rates bounded away from zero. For this analysis, we first consider the model without inhibition and discuss the impact of the refractory period on the neuron’s output rate. We then discuss the impact of inhibitory inputs.

Formulation of output rate as dead-time modified Poisson process. In our simulations, we typically use excitatory inputs that are sufficiently strong so that, in the absence of refractory effects and inhibition, a single excitatory input event triggers an output spike. In response to weak excitatory inputs, the model produces low spike rates that are abolished by modest amounts of A-current, so we do not analyze the weak-input regime. At low input rates in the strong-input regime and with g_Syn,I = 0, we expect generation of output spikes to replicate the sequence of input excitatory events. Specifically, the output spike train will follow a homogeneous Poisson process with the same rate as excitatory events: r_out = r_E.

For high input rates, a refractory effect prevents the neuron from firing on a one-to-one basis with each input event. A simple approximation of the input/output relationship in this “high input rate” regime is to assume there is a fixed period of duration R ms during which the neuron cannot fire, and therefore we say (11) Since input events are drawn from a homogeneous Poisson process, the output events under this approximation follow a dead time modified Poisson process with dead time (i.e. refractory period) R. The input/output firing rate relationship is then given by (12) where rate is defined as the expected number of events a time interval, divided by the duration of that interval [26].

Definition of the firing threshold. The approximation of r_out introduced in Eq 12 does not incorporate any effect of inhibition. To incorporate the effects of inhibition into the formulation, we presume that excitatory input events will produce output spikes (following the dead-time modified Poisson process, as described above) unless the excitatory event occurs at a time when synaptic inhibition is sufficiently strong to prevent spiking. In other words, we define a firing threshold θ = θ(r_E, g_A) to be the value of the inhibitory conductance s_I for which an excitatory input will evoke a spike only if s_I < θ. For now, we assume that θ is known. Later, we describe how θ can be computed.

Suppose that inhibitory events occur periodically with period P_I. At the onset of each inhibitory event, the inhibition conductance s_I increases to 1 and then exponentially decays with time constant 1/β_I. It follows that the fraction of time that that s_I < θ is given by (13) By the definition of the firing threshold, this is the fraction of time that the neuron responds to excitatory input. Hence, we modify the Poisson approximation of output firing rate to be (14) To complete the analysis, it still remains to compute the firing threshold, θ, which depends on the parameters r_E and g_A.

Computing the firing threshold. Our derivation of the firing threshold procedes as follows. Let n_av be the average value of {n_k: k = 1, 2,….} such that for each k: i) there is an excitatory input at some time, t₀, with n(t₀) = n_k; and ii) the neuron responds to this excitatory input with a spike. We show below that if n_av is known, then we can compute the firing threshold, θ, while if θ is known, then we can compute n_av. This gives rise to two maps: θ = Θ(n_av) and n_av = N(θ). If (θ*, n*) is a fixed point of these two maps, then the firing threshold is given by θ = θ*. In what follows, we make several simplifying assumptions. We compare predictions of the analysis with simulations of the reduced model later.

Suppose that n_av is given. Here we assume that if an excitatory input arrives at time t₀ and the neuron responds to this input with a spike, then n(t₀) = n_av. Recall that the neuron will spike in response to an excitatory input at time t₀ if (V(t₀), n(t₀)) lies below the left knee of s_E = 1 nullcline, which depends on the other variables, b and s_I. As before, let . The firing threshold can then be defined as the value of s_I so that N_lk(s_I, r_E, g_A) = n_av. This defines the map θ = Θ(n_av).

Now suppose we are somehow given the firing threshold θ and wish to estimate n_av. The first step is to determine the position of the right knee of the s_E = 1 cubic, which we denote as N_rk. This depends on s_I, b and g_A; however, as the dependence of N_rk on each of these variables is weak, we will assume that N_rk is a constant. We determine its value using XPPAUT.

We next consider the evolution of n in the silent phase. Here, we assume that n_∞(V) = 0 and τ_n(V) = τ₀, a constant. Then, while in the silent phase, n(t) satisfies , so that If the firing threshold θ is known, then the output firing rate, r_out, is given by Eq 14. The average interspike interval is, therefore, 1000/r_out ms, which we assume is the average time the neuron spends in the silent phase. In this case, the average value of n is given by This defines the map n_av = N(θ).

We have now defined the two maps, θ = Θ(n_av) and n_av = N(θ). By taking the composition of these maps, we obtain the fixed point problem: n = N ∘ Θ(n). This map is continuous in n, and n takes values in the closed interval [0, 1], so there exists a fixed-point of this map [27]. We use successive iterations of the bisection method to numerically compute n_av as the solution of the fixed point fixed point problem n = N ∘ Θ(n), and then set θ = Θ(n_av).

Slope of output rates at onset of firing in the divisive case. By computing the firing threshold at r_E = 0, we can study the behavior of the model near the onset of firing. In particular, if g_A is sufficiently small or g_Syn,E is sufficiently large, then inhibition is divisive and the neuron is able to fire in response to arbitrarily low firing rates (recall Fig 2). Our goal here is to compute the slopes of the input/output firing rate curves at r_E = 0 in the case of divisive inhibition.

This is done by simply differentiating the firing rate approximation in Eq 14 with respect to r_E and setting r_E = 0. This yields (15) where θ₀ is the firing threshold when r_E = 0. This can be easily computed using XPPAUT as follows. Since we are considering arbitrarily low firing rates, it follows that the average value of n at times of output spikes is n_av ≈ 0, where n_av was defined in the preceding section. The analysis in that section demonstrates that θ₀ = Θ(0), i.e. it is the value of s_I so that N_lk(s_I, 0, g_A) = 0.

In Fig 9, we plot the slope of the input/output firing rate curves at r_E = 0 computed from both the theoretical prediction (Eq 15) and simulations of the full model. There is a tendency for the approximated value of the slope at firing onset to overestimate the slope computed in simulations of the full model. Nonetheless, the theory captures qualitative features of the relationship between slope at firing rate onset and A-channel conductance. In particular, the slope at firing rate onset decreases as g_A increases, indicating gain control by inhibition is “stronger” with higher values of g_A. The value of g_A at which the slope reaches zero (g_A between 25 and 30) was denoted as previously; it is the critical value of g_A at which the effect of inhibition switches from divisive to subtractive.

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Fig 9. The slope of the input/output firing rate curves at r_E = 0 computed from both the theoretical prediction Eq 15 and simulations of the full model.

https://doi.org/10.1371/journal.pcbi.1006292.g009

Approximation of output rates at arbitrary input rates. To more completely characterize output firing rates, we seek to extend our approximations to cases of higher output rates. In other words, we consider the firing rate equation (Eq 14) for r_E > 0. We previously described how to compute θ. With θ known, the only undetermined parameter in the firing rate equation is R. We interpret this parameter as the duration of the refractory period in the model. The value of R depends on the internal dynamics of the model, strength of excitatory inputs, and other factors. To obtain approximations for R, we simulated firing rate input/output curves (without inhibition) and performed a nonlinear curve-fit using Eq 12 to estimate its value. We find that R ≈ 10 ms (with some dependence on g_a) provides accurate approximations to the firing rate functions in the absence of inhibition.

We compare our theoretical approximation to the firing rate of the model neuron to simulated firing rates in Fig 10. The dead-time modified Poisson process provides a satisfactory description of firing rate responses without inhibition (black, top row), for various values of g_A (values of g_A increase from 15 to 35, in increments of 10, from left-to-right in this figure). When we include inhibition in these simulations, we find that the approximated firing rate (using the firing rate threshold computation outlined above) tends to overestimate simulated firing rates (colors). Nevertheless, approximations do capture the qualitative differences in firing rate curves for divisive inhibition (smaller values of g_A) and subtractive inhibition (larger values of g_A).

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Fig 10. Output firing rates approximated as dead-time modified Poisson process with firing threshold.

Top row: Firing rate as a function of input rate obtained from simulations (circles) and theoretical approximation (lines); for g_A = 15 (A1), g_A = 25 (B1), and g_A = 35 (C1). Simulations and theory show transition from divisive to subtractive inhibition as g_A increases. Bottom row: Theoretical approximation of firing threshold θ (lines), and the largest observed values of s_I for which excitatory inputs elicited spikes in simulations (circles), plotted as functions of input rate; for g_A = 15 (A2), g_A = 25 (B2), and g_A = 35 (C2).

https://doi.org/10.1371/journal.pcbi.1006292.g010

The firing threshold calculation defines θ, the theoretical value of s_I for which the neuron cannot produce a spike if an incoming excitatory event arrives at a time when s_I ≥ θ. We plot values of θ as colored lines in the lower row of Fig 10. To compare θ to simulations, we recorded the values of s_I at the time of every excitatory event that triggered a spike. Then, for each input rate (x-axis), we found the maximum value of s_I for these spike-triggering excitatory events. These maximum s_I values qualitatively align with our approximations of θ, further supporting the heuristic concept of a firing threshold.

We point out that, following from Eqs 12 and 14 that a “purely divisive” response would be one for which the firing threshold is constant with respect the input rate. This can (nearly) occur for specific parameter sets; see for instance the nearly constant threshold in Fig 10A2. However, firing threshold depends in general on input rate. Furthermore, a “purely subtractive” response would be one for which the firing threshold is piecewise constant (a step function). The firing threshold in Fig 10C2 increases with input rate in a non-monotonic fashion, so inhibition in this case, in fact, has both divisive and subtractive characteristics. This is true in general and we reiterate that our use of the term “subtractive” encompasses cases for which inhibition has both divisive and subtractive effects.

Results for non-instantaneous A-current activation

A key assumption in the formulation and analysis of the reduced model is that the A-current activates sufficiently fast so that the dynamical variable a can be set to its voltage-dependent steady state value; that is, we set a = a_∞(V). One effect of this change from a evolving dynamically with τ_a = 2 ms (the full model), to a evolving instantaneously as a_∞(V) (the reduced model), is that excitation must be much stronger in the reduced model to observe subtractive inhibition. Typical values of g_Syn,E in the full model are around 0.5 (see Fig 4), and typical values of g_Syn,E in the reduced model are around 3. This suggests that the speed of A-current activation (not just the strength of the A-current) plays a role in switching the effect of inhibition from divisive to subtractive.

For inhibition to have a subtractive effect, responses to infrequent excitatory inputs must be suppressed. In the reduced model, this occurs when g_A is sufficiently strong because the A-type channel activates “instantaneously” and can prevent spike initiation. In the one-compartment model with “non-instantaneous” a variable, large g_A could switch the effect of inhibition to subtractive, but only if excitatory input strength was also sufficiently small (recall Fig 4). The importance of small g_Syn,E is demonstrated in Fig 11. We show time-courses of voltage in the one-compartment model for g_Syn,E = 0.2, 0.5, and 1, and with g_A = 0 and g_I = 0. In all cases, the input evokes an output spike. Notice, however, that as input strength weakens, there is a marked delay in the time before spike initiation. For the weakest input used (g_Syn,E = 0.2), there is a delay of roughly 2 ms before the rapid upstroke of V at the onset of the action potential. During this “pause”, voltage is slowly ramping up and, simultaneously, recruiting additional A-current as the a variable activates. The amount of I_A available to suppress spike initiation depends, therefore, on A-channel maximal conductance (g_A) and also the time-constant of I_A activation (τ_A).

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Fig 11. Role of τ_A in determining switch between divisive and subtractive inhibition.

A: Voltage traces in response to excitatory inputs of varying strengths (g_A = 0, and g_Syn,I = 0). B: Threshold-linear relation between output firing rates in simulations of one-compartment model with and without inhibition for varying A-channel activation time constant (τ_A = 0.5, 1, 2) and g_A = 20. Synaptic parameter values are g_Syn,E = 0.5, and (for simulations with inhibition) g_Syn,I = 1 and r_I = 50. C: Critical values of g_A that define boundary between subtractive and divisive inhibition in (g_Syn,E, g_A) parameter space. The boundary shifts downward as τ_A decreases, indicating that faster activating A-current enables inhibition to have a subtractive effect for lower values of g_A.

https://doi.org/10.1371/journal.pcbi.1006292.g011

From this observation, we draw the following conclusion: “non-instantaneous” I_A can act to switch the effect of inhibition from divisive to subtractive, but only if it activates rapidly enough relative to the dynamics of spike initiation. To illustrate our point, we simulated the model with three values of A-channel activation time constant (τ_A = 0.5, 1, 2), using g_Syn,E = 0.5, g_Syn,I = 1, and r_I = 50 Hz. As shown in Fig 11B, inhibition is divisive for slower activation kinetics (τ_A = 1, 2) and subtractive for faster activation kinetics (τ_A = 0.5). In Fig 11C, we map the boundary between divisive and subtractive inhibition in the (g_Syn,E, g_A) parameter plane. There is a strong effect of τ_A. Faster activation kinetics (smaller τ_A values) shift the critical point at which inhibition switches from divisive to subtractive to lower values values of g_A.

Results for multi-compartment model

Our prior observation, that delaying spike initiation allows inhibition to have a subtractive effect for “non-instantaneous” A-channel activation, led us to investigate other cellular mechanisms that could have a similar effect. To this end, we considered a multi-compartment neuron model that describes a soma and passive dendrite. Inhibition and voltage-gated currents are restricted to the soma compartment, and excitation targets a location somewhere on the dendrite. Passive cable theory tells us that the amplitudes of excitatory post-synaptic potentials attenuate and their rising slopes become less steep as signals spread along the cable [28]. By varying the location of excitatory synaptic inputs to the dendrite in the multi-compartment model, we can, therefore, adjust the shape of excitatory post-synaptic potentials as they arrive in the soma.

Examples of action potentials, recorded in the soma compartment, in response to inputs at different locations along the dendrite are shown in Fig 12A. Synaptic conductance strength is constant (g_Syn,E = 2 in these simulations). Inputs that arrive proximal to the soma are large and fast-rising relative to responses to more distal inputs, and thus evoke action potentials with shorter latencies. The parameter cpt_in identifies the compartment that receives synaptic excitation. It takes values from 1 (proximal) to 9 (distal).

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Fig 12. Divisive and subtractive inhibition in a multi-compartment neuron model.

A: Voltage traces in response to excitatory inputs at varying input locations along the dendrite. Parameter values in these simulations: g_Syn,E = 3, g_Syn,I = 0, and g_A = 0. Inputs distant from the soma lead to spike initiation with millisecond-scale delay between excitatory input and spike onset. B: Threshold-linear relation between output firing rates in simulations of the multi-compartment model with and without inhibition for varying input location and g_A = 20. For simulations with inhibition: g_Syn,I = 1 and r_I = 50. Inhibition is subtractive for distal excitatory input (cpt_in = 6). C: Critical values of g_A that define boundary between subtractive and divisive inhibition in (g_Syn,E, g_A) parameter space. The boundary shifts downward as excitatory inputs are moved to more distal locations, indicating that inhibition has a subtractive effect for lower values of g_A for more distal inputs.

https://doi.org/10.1371/journal.pcbi.1006292.g012

We included synaptic inhibition in the model targeting the soma, and used simulations to characterize the effect of inhibition as either subtractive or divisive. In Fig 12B we observe a transition from divisive to subtractive inhibition as we move the location on the dendrite at which synaptic excitation targets the cell. This matches our expectation: synaptic excitation placed at more distant locations will generate weaker and slower rising inputs in the soma. This will, in turn, lead to spikes that initiate slowly and that give time for A-channel conductance to activate and prevent spike generation. In additional simulations included as S4 Fig we targeted inhibitory inputs to the dendrite and did not observe substantially different results.

We explored the (g_A, g_Syn,E) parameter plane and identified the boundary separating regions in which inhibition has a divisive effect on firing rates and regions in which inhibition has a subtractive effect (following the procedure used previously for Fig 4). We find that there is a dramatic effect of input location, as shown in Fig 12C. The region of the (g_A, g_Syn,E) parameter plane over which inhibition has a divisive effect is smaller when inputs are more distant from the soma.

Analysis identifies when I_A promotes divisive or subtractive inhibition

Using simulations and phase plane analysis, we systematically investigated conditions under which inhibition acts on firing rate outputs in a divisive or subtractive manner. We first identified critical values of I_A conductance (g_A) at which the effect of inhibition switched from divisive (for lower g_A values) to subtractive (for higher g_A values) (Fig 4). In the reduced model, we approximated this critical value of g_A using bifurcation analysis. By tracking the left-knee of the V-nullcline (Fig 5), we identified this critical value of g_A as a bifurcation point at which the neuron model ceased to be excitable in response to synaptic inputs (Fig 8). Key in this analysis was the separation of time scales between fast variables (V) and slow variables (n, b). In fact, the inactivation variable b was sufficiently slow that it could be treated as a constant with a value that depended on the input rate (Fig 7). This simplification enabled further analysis. By viewing the spiking output of the model as a Poisson process modified by a refractory period and inhibition-dependent firing threshold, we approximated firing rates at the point of spiking onset (Fig 9) as well as for arbitrary input rates (Fig 10).

A-type potassium current is a source of dynamic, voltage-gated negative feedback that is fast activating. We leveraged this property to obtain analytical results by assuming that the gating variable for I_A activation, a, evolved instantaneously to its voltage-dependent equilibrium value (see also [23]). We also performed simulations without this assumption and discovered a delicate interaction between the speeds of I_A activation and spike initiation. In particular, subtractive inhibition required that I_A is sufficiently strong and that it activates sufficiently rapidly to prevent spike initiation (Fig 11). For our standard value of I_A activation (τ_A = 2 ms), we found that, in conditions of slow spike initiation, I_A could “ramp up” during slowly-developing spikes and suppress spike initiation. Weak excitatory inputs, or excitatory inputs targeting more distal regions in a model that included a spatially-extended dendritic process (Fig 12) produced spikes that were slow to initiate, and were therefore scenarios in which inhibition was subtractive for low to modest levels of g_A.

Relation to previous works

Divisive inhibition is a mechanism of neural gain control and has been the subject of numerous studies; see [1] for review. We found that, at lower levels of g_A, inhibition can have a divisive effect on the input/ouput properties of a spiking neuron responding to a mixture of random excitatory inputs and periodic inhibitory inputs. The amount of g_A altered the slope (gain) of the output firing rate, and thereby tunes the gain control in this system. This result is consistent with the results of a recent in vitro study of neurons in the rostral nucleus of the solitary tract [24]. In that experiment, Chen and colleagues controlled inhibition using optogenetic techniques and constructed threshold linear functions to express the relation between firing responses with and without inhibition (analogous to our Fig 2C, and similar figures). They observed that slopes of threshold-linear function were more shallow for neurons with I_A, as compared to neurons in the same nucleus that did not have I_A. Thus, the presence of I_A enhanced the divisive effect of inhibition. Previous modeling work has identified similar gain control effects by I_A [29].

We observed, additionally, that at higher levels of g_A, the A-type current can switch the effect of inhibition from divisive to subtractive. This demonstrates a novel example of how the internal dynamics of a neuron interact with synaptic inhibition to change the neuron’s computational properties (input/output relation). Previous studies have explored the multi-faceted ways in which I_A current can alter neural dynamics. Connor and Stevens established I_A current as a mechanism to prolong interspike intervals of repetitively-firing neurons to arbitrary lengths (“type I” firing dynamics) [11]. Other identified functions of I_A include prolonging first spike latency [30], producing burst firing patterns and preventing anodal break (rebound) firing [23], filtering synaptic inputs in favor of slow time-scale NMDA receptor-mediated inputs [31], and affecting the correlation in spiking among neurons responding to common inputs [32]. Our contribution adds to the rich repertoire of I_A function.

Implications for modulation of inhibitory effects

We have identified routes to subtractive inhibition that depends only on mechanisms that could be readily adjusted by processes of plasticity and neuromodulation. In particular, we have shown that strong and fast I_A can lead to subtractive inhibition. The strength and kinetics of the A-type Potassium channels can be modulated in a variety of ways [33–36]. For example, in neurons involved in gastro-intestinal function, A-type potassium channels were modified both by diet [37–39] and gastric disorders [40].

We also found that weak excitatory inputs or more distally-located excitatory inputs led to subtractive inhibition by slowing the onset of action potentials. Synaptic plasticity and modulation of the electrical properties of dendrites can adjust the strength and propagation of excitatory inputs [41], and plasticity of spike initiation zones could change the dynamics of spike initiation [42, 43]. These changes happen at the level of the output neuron. They do not require “global” modulatory effects to change background network activity, circuit structure, or the balance of excitation and inhibition. We conclude, then, that I_A can add flexibility to neural systems by allowing neurons to “self-regulate” whether inhibition acts in a subtractive or divisive manner.