### Model

We consider the dynamics of an LIF model neuron with excitatory synaptic inputs as governed by the equations

{v}^{\prime}=I-v-g(v-E),

(2)

{g}^{\prime}=-\beta g,

(3)

together with the reset condition

v({t}^{-})={v}_{th}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}v({t}^{+})={v}_{r}.

(4)

Equation (2) can be derived from a conductance-based equation C{v}^{\prime}={I}_{\mathit{input}}-{\sum}_{j}{g}_{j}(V-{E}_{j})-{g}_{\mathit{syn}}(V-{E}_{\mathit{syn}}) with fixed intrinsic current conductances {g}_{j}, but we think of it as a nondimensionalized abstract model in which voltage intrinsically converges to a baseline *I* and *E* is the reversal potential of a synaptic input with strength g>0. We assume {v}_{r}<I<{v}_{th}, such that no spikes are fired in the absence of input, and E>{v}_{th}, and we consider the invariant half-plane \{g\ge 0\} within the (v,g) phase plane, where (I,0) is the unique stable critical point. Further, we represent the excitatory input by the equations

g({t}_{n}^{-})=g({t}_{n}^{+})+{k}_{n},

(5)

\sum _{n=1}^{N}{k}_{n}=G

(6)

for {k}_{n}\in (0,G], n=1,2,\dots, with N\ge 1 finite and G>0 fixed in **R**. Equations (3) and (5) show that each input kick can be chosen to arrive at any time and instantaneously updates the value of *g* when it arrives and that the synaptic conductance *g* always decays exponentially between kicks. Equation (6) states that the sum of all inputs, however they may be divided, is always equal to a fixed input allowance *G*. In the subsequent subsections, we assume that *I*, *E*, {v}_{th}, {v}_{r}, and *β* are fixed and consider how to partition and time the input *G* to yield the greatest number of threshold crossings, or spikes. Without loss of generality, we take {v}_{r}=0. First, we discuss a phase plane representation of this problem and consider some example strategies.

### Phase plane structures and basic strategies

We illustrate some key structures in the phase plane for system (2), (3) in Figure 1. The *v*-nullcline, based on equation (2), is the curve \{g=(v-I)/(E-v)\}. Denote the trajectory through the point {\Gamma}_{0}^{+}:=({v}_{th},{g}_{0}^{+}=({v}_{th}-I)/(E-{v}_{th})) on this curve by Γ_{0}, which is tangent to the line \{v={v}_{th}\}. Γ_{0} partitions the set \{(v,g):v<{v}_{th},g\ge 0\} into a set of initial conditions that yield spikes, namely those above Γ_{0}, and a set of initial conditions that do not yield spikes, namely those below Γ_{0}. Let {\Gamma}_{0}^{-} denote the intersection of Γ_{0} with \{v={v}_{r}\} and let {g}_{0}^{-} denote its *g* coordinate. The point {\Gamma}_{1}^{+}:=({v}_{th},{g}_{0}^{-}) is on the threshold line, and there is a trajectory Γ_{1} that flows through this point. The band between Γ_{0} and Γ_{1}, with {v}_{r}\le v\le {v}_{th}, consists of the set of initial conditions from which trajectories yield exactly one spike. Denote this band by {B}_{1}. Similarly, for each natural number *n*, we can define a curve {\Gamma}_{n}, such that each trajectory with an initial condition in the band {B}_{n} between {\Gamma}_{n-1} and {\Gamma}_{n} yields *n* spikes. Note that this band structure exists if parameters are altered such that I>{v}_{th}, although the minimal band is shifted to negative *g* values, and hence the methods that we discuss easily generalize to this case.

The band structure that characterizes sets of initial conditions that yield particular numbers of spikes calls to mind several natural strategies for doling out input kicks to maximize spike output:

#### Critical kicks

Once a trajectory is below Γ_{0}, it approaches (I,0) asymptotically. Let {\gamma}_{c} denote the *g*-coordinate of the intersection \{v=I\}\cap {\Gamma}_{0}. One possible strategy is to give an initial input of size {\gamma}_{c} as well as subsequent inputs of size {\gamma}_{c} each time the trajectory reaches a sufficiently small neighborhood of (I,0); see Figure 1. This *critical kicks* strategy yields *n* spikes where n{\gamma}_{c}\le G<(n+1){\gamma}_{c}.

#### Big kick

A possible disadvantage of the critical kicks strategy is that the increment {\gamma}_{c} to achieve a spike may exceed the width {\gamma}_{n} between bands {\Gamma}_{n} and {\Gamma}_{n-1} for n\ge 1. To ensure that this increment is only encountered once, taking inspiration from the power of synchronized inputs [12, 13], another reasonable strategy is to give a single *big kick* of size *G*, all at once, to the resting cell (Figure 1).

#### Reset and kick

An input of size {\gamma}_{c} is sufficient to push the voltage across the threshold for single spike initiation. In the big kick strategy, the additional available input G-{\gamma}_{c} is provided together with the {\gamma}_{c}. It is possible that this additional input could deliver more spikes if it were delivered separately from the initial {\gamma}_{c}. We can define a strategy, for example, in which an initial kick of size {\gamma}_{c} is given to elicit a spike. As soon as this spike is fired, the cell is reset and the remaining input allowance G-{\gamma}_{c} is given. Clearly, this *reset and kick* strategy would make sense if the bands {B}_{n} were narrowest at v=0.

#### Threshold kick

At the other extreme, the bands {B}_{n} might be narrowest at v={v}_{th}. In this case, a possible optimal strategy would be the *threshold kick* strategy, defined by giving the initial kick of size {\gamma}_{c} and following this with a kick of size G-{\gamma}_{c} just before threshold crossing occurs (Figure 1).

Intuitively, it is reasonable to think that if *β* is large, such that inputs rapidly decay, then it makes sense to dole out inputs in minimal pieces, such that something like the critical kicks strategy may be optimal. Alternatively, if *β* is small, such that inputs decay slowly, then it makes sense to make inputs available as early as possible, such that one of other strategies is likely to be optimal. To analyze more carefully which strategy is optimal, it will be helpful to define a *band width*
{\delta}_{n}(v) as the distance from {\Gamma}_{n-1} to {\Gamma}_{n} in the *g*-direction for each fixed v\in [0,{v}_{th}]. With this definition in hand, we note that {\delta}_{n+1}({v}_{th})={\delta}_{n}({v}_{r}) for n\ge 1, and hence the reset and kick and threshold kick strategies are effectively the same strategy, yielding the same number of spikes (Figure 1). We also let {\delta}_{\infty}(v)={lim}_{n\to \infty}{\delta}_{n}(v), v\in [0,{v}_{th}], if this limit exists. Very roughly speaking, the critical kicks strategy will yield approximately G/{\gamma}_{c} spikes while the other strategies will induce about G/{\delta}_{\infty}(v) spikes for some *v*, at least if {\delta}_{n}(v) converges to {\delta}_{\infty}(v) quickly. Thus, comparison of {\gamma}_{c} and {\delta}_{\infty} can be used to give an initial suggestion of what strategy to follow.

The value of {\gamma}_{c} can be observed numerically by backwards integration from ({v}_{th},{g}_{0}^{+}) until v=I. Alternatively, it may be optimal to replace {\gamma}_{c} by the distance {\tilde{\gamma}}_{c}:={g}_{0}^{-}-{g}_{0}^{+}, as can be computed by backwards integration from ({v}_{th},{g}_{0}^{+}) up to ({v}_{r},{g}_{0}^{-}), and give kicks of size {\tilde{\gamma}}_{c} after each spike, after the initial kick of size {\gamma}_{c}; we will still refer to this as a critical kicks strategy.

An approximate value of {\delta}_{\infty}(v) can be derived as follows. From (2), (3), the slope *s* of the vector field at any point in the phase plane is given by

s(v,g)=\frac{-\beta g}{I-v-g(v-E)}.

(7)

For v\in [{v}_{r},{v}_{th}],

{s}^{\ast}(v):=\underset{g\to \infty}{lim}s(v,g)=\frac{-\beta}{E-v}<0.

The magnitude of the change in *g* over one spike cycle is

\begin{array}{rl}{\delta}_{\infty}:=& {\int}_{{v}_{th}}^{{v}_{r}}{s}_{v}^{\ast}\phantom{\rule{0.2em}{0ex}}dv\\ =& {\int}_{{v}_{th}}^{{v}_{r}}\frac{-\beta}{E-v}\phantom{\rule{0.2em}{0ex}}dv=\beta ln\left|\frac{{v}_{r}-E}{{v}_{th}-E}\right|.\end{array}

(8)

By construction, {\delta}_{\infty}({v}_{r})={\delta}_{\infty} as given by (8). But since the value of *g* at reset for the trajectory forming the upper bound of one band is the value of *g* at threshold for the trajectory forming its lower bound, and we have taken the limit as n\to \infty, we can also estimate {\delta}_{\infty}({v}_{th}) using (8) and indeed, using similar translation arguments, we estimate {\delta}_{\infty}(v)={\delta}_{\infty} for all v\in [{v}_{r},{v}_{th}].

Comparison of {\gamma}_{c} (or {\tilde{\gamma}}_{c}) and {\delta}_{\infty} suggests whether or not the critical kicks strategy will elicit more spikes than the other strategies we have described. If not, then we need additional arguments to assess the relative effectiveness of these alternative strategies. In fact, regarding alternative strategies, we have the following result:

**Proposition 1** *The big kick strategy always yields at least as many spikes as the reset and kick* (*and equivalently*, *the threshold kick*) *strategy*.

*Proof* The reset and kick strategy yields m+1 spikes, where *m* is the largest integer such that

\sum _{n=1}^{m}{\delta}_{n}({v}_{th})\le G-{\gamma}_{c}.

Using equation (7), compute

\frac{ds}{dg}=\frac{-\beta (I-v)}{{(I-v-g(v-E))}^{2}}.

(9)

We can see from equation (9) that if v<I, then the slope *s* becomes more negative as *g* is increased, and if v>I, then the slope *s* becomes less negative as *g* is increased. Thus, the bands are narrowest at v=I; that is, {\delta}_{n}(I)<min\{{\delta}_{n}({v}_{r}),{\delta}_{n}({v}_{th})={\delta}_{n+1}({v}_{r})\}. Hence, the big kick strategy, which elicits {m}_{b}+1 spikes for the largest integer {m}_{b} such that

\sum _{n=1}^{{m}_{b}}{\delta}_{n}(I)\le G-{\gamma}_{c},

always generates at least as many spikes as reset and kick. □

### Band width estimation

In the previous subsection, we introduced a small number of intuitively reasonable strategies for eliciting the maximum number of spikes from model (2), (3) using a constrained input. We also used a phase plane approach to define a natural band structure, along with a corresponding idea of a band width {\delta}_{n}(v), which we used to show that two of these, the reset and kick and threshold kick strategies, will never be optimal. This structure can also be used to obtain an intuitive idea of which conditions favor a big kick strategy and which conditions favor the critical kicks strategy of giving many small kicks of the same particular size. Next, we use some approximations to derive additional quantitative information about {\delta}_{n} that can be used to determine more globally the optimal input strategy. Henceforth, in addition to assuming that {v}_{r}=0, we for convenience set {v}_{th}=1, with 0<I<1<E.

We can estimate the magnitude \delta (g) of the change in *g* that occurs over one spike cycle using the slope s(v,g) given in equation (7),

\begin{array}{rl}\delta (g)& ={\int}_{1}^{0}\frac{-\beta g}{I-v-g(v-E)}\phantom{\rule{0.2em}{0ex}}dv\\ \approx -\frac{\beta g}{1+g}ln\frac{I+gE-1-g}{I+gE},\end{array}

(10)

where we have approximated *g* by a constant to estimate the integral. Note that for each *n*, the band widths {\delta}_{n}(0)={\delta}_{n+1}(1) from the previous subsection are approximately equal to \delta (g) for certain corresponding choices of *g*; for example, {\delta}_{1}(0)\approx \delta (({g}_{0}^{-}+{g}_{1}^{-})/2), where Γ_{1} intersects \{v=0\} at (0,{g}_{1}^{-}). More generally, it is not necessary to choose a *g* associated with the boundary of a band, as defined from the previous subsection, in order to compute \delta (g).

We can investigate the spikes of a cell by analyzing (10). This approach yields the following result.

**Proposition 2** *If*
E+I-2EI\ge 0, *then*
\delta (g)
*is a monotone decreasing function of g*. *If*
E+I-2EI<0, *then*
\delta (g)
*has a unique local minimum at a finite*, *positive value*
g={g}_{0}.

*Proof* Calculating the derivative of \delta (g) with respect to *g*, we have

\frac{d\delta (g)}{dg}=\frac{\beta}{{(1+g)}^{2}}f(g),

where

f(g):=ln\left(\frac{I+gE}{I+gE-1-g}\right)-\frac{g(E-I)(1+g)}{(I+gE-1-g)(I+gE)}.

Furthermore,

\frac{df(g)}{dg}=\frac{(E-I)(1+g)(g(E+I-2EI)+2I(1-I))}{{(I+gE-1-g)}^{2}{(I+gE)}^{2}}.

(11)

Equation (11) shows that E+I-2EI is indeed a key quantity.

Suppose now that E+I-2EI\ge 0. If E+I-2EI\ge 0, then df(g)/dg\ge 0, such that f(g) increases as *g* increases. Define

{f}_{1}\equiv \underset{g\to +\infty}{lim\hspace{0.17em}inf}f(g)=ln\frac{E}{E-1}-\frac{E-I}{E(E-1)}.

Since

\frac{d{f}_{1}}{dE}=\frac{E+I-2EI}{{E}^{2}{(E-1)}^{2}}\ge 0,

(12)

increases as *E* increases. As

\underset{E\to +\infty}{lim\hspace{0.17em}inf}{f}_{1}=0,

we have {f}_{1}\le 0. Therefore, f(g)\le 0 for all *g* and hence d\delta (g)/dg<0. In another words, under the original approximation used to obtain (10), \delta (g) decreases, and thus there is less change in *g* across each cycle from reset to threshold, as *g* increases.

Next, suppose that E+I-2EI<0. Under this condition, df/dg changes signs, with df/dg>0 for g\in (\frac{1-I}{E-1},\frac{2I(1-I)}{2EI-E-I}) and df(g)/dg<0 for g\in (\frac{2I(1-I)}{2EI-E-I},+\infty ). From expression (12), we have d{f}_{1}/dE\le 0 and {lim\hspace{0.17em}inf}_{E\to +\infty}{f}_{1}=0, such that {f}_{1}\ge 0 for all *E* and f(g)\ge 0 for g\in (\frac{2I(1-I)}{2EI-E-I},+\infty ). Thus, there exists a unique point {g}_{0} such that f(g)\le 0 for all g\in (\frac{1-I}{E-1},{g}_{0}) and f(g)\ge 0 for all g\in ({g}_{0},+\infty ). Correspondingly, \delta (g) decreases when g\in (\frac{1-I}{E-1},{g}_{0}) and increases when g\in ({g}_{0},+\infty ), where {g}_{0} is the zero point of f(g)=0, and the proof is complete. □

In the case of E+I-2EI\ge 0, the monotonicity of \delta (g) suggests that for a trajectory evolving from an initial condition of (v,g)=(0,g(0)) to a final condition (v,g)=(1,g(t)), the drop g(0)-g(t) should be smaller for larger g(0). From Figure 2, we can see that, while the approximation used to derive equation (10) introduces an error in relative to the actual change in *g* computed from direct simulation of trajectories with E+I-2EI\ge 0, the error appears to be small and the monotonicity of \delta (g) appears to be correct. Similarly, a numerically computed example of \delta (g) for parameters that yield E+I-2EI<0 is shown in Figure 3.

### Optimal strategies and spike counts

Based on the previous two subsections, we conclude that if E+I-2EI\ge 0, then the loss of input with each spike, measured by \delta (g), decreases as *g* increases and, furthermore, an input of a fixed size will cross the most spiking bands if it is given at v=I. Hence, of all possible strategies for eliciting spikes with an input of total size *G*, the one that yields the most spikes is what we earlier called the big kick strategy, unless {\gamma}_{c} (or {\tilde{\gamma}}_{c}) is sufficiently small that avoiding the bands altogether by following the critical kicks strategy is optimal. The optimal strategy when E+I-2EI<0 is to provide a kick that puts *g* at approximately {g}_{0}, the *g* value where the minimum of \delta (g) occurs, and then provide as many kicks as possible of size \delta ({g}_{0}), again assuming that {\gamma}_{c},{\tilde{\gamma}}_{c} are above a certain size. We will next perform some additional calculations that can provide estimates of numbers of spikes resulting from any strategy, which can be used with a minimum of calculation to compare the results of particular input sequences.

We first suppose that E+I-2EI<0 and consider a generalization of the optimal strategy described above. That is, we assume that an initial input {G}_{i}\ge {g}_{0} is given and then, once *g* evolves to some neighborhood of {g}_{0}, kicks of size \delta ({g}_{0}) are repeatedly applied until the remaining input falls below \delta ({g}_{0}). We now estimate the numbers of spikes Ω fired for each strategy of this type. Let Ω_{1} denote the number of spikes fired during the initial time period when *g* drops toward {g}_{0}, let Ω_{2} denote the number of spikes fired during the final time period after the available input is depleted, and let Ω_{3} denote the number of spikes fired during the intervening period when repeated kicks of size \delta ({g}_{0}) are given. Clearly,

{\Omega}_{3}\approx \frac{G-{G}_{i}}{\delta ({g}_{0})}

(13)

and \Omega ={\Omega}_{1}+{\Omega}_{2}+{\Omega}_{3}, so it remains to estimate Ω_{1} and Ω_{2}.

Because of the shape of the function \delta (g), the largest \delta (g) during the initial time period will be associated with the first spike fired, while the largest during the final period will be associated with the last spike fired. We can estimate the drop \delta ({G}_{i}) in *g* up to the firing of the first spike from equation (10). To make this estimate relevant to other spikes early in the spike train, we take v(0)=0 rather than v(0)=I. Approximating the level of *g* in equation (10) by {g}_{i}:={G}_{i}-\delta ({G}_{i})/2, we obtain

\delta ({G}_{i})\approx {\delta}_{i}:=-\frac{\beta {g}_{i}}{1+{g}_{i}}ln\frac{I-1+(E-1){g}_{i}}{I+E{g}_{i}}.

(14)

Next, we estimate \delta (g) for the final spike fired, which we call {\delta}_{f}. To do this, we assume that when the final spike is fired, the trajectory reaches the lower bound on the *g* values that can yield a spike, namely the point of intersection of the *v*-nullcline and \{v=1\}, at which g={g}_{0}^{+}=(1-I)/(E-1). We also use the intermediate value of *g* across the trajectory {\Gamma}_{0}(v), namely (1-I)/(E-1)+{\delta}_{f}/2, as the value of *g* for equation (10), which yields

\begin{array}{rl}{\delta}_{f}=& -\frac{\beta [{\delta}_{f}/2+(1-I)/(E-1)]}{1+{\delta}_{f}/2+(1-I)/(E-1)}\\ \times ln\frac{I-1+(E-1)[{\delta}_{f}/2+(1-I)/(E-1)]}{I+E[{\delta}_{f}/2+(1-I)/(E-1)]}.\end{array}

(15)

Now, to obtain an estimated spike count as *g* decays from {G}_{i} to {g}_{0}, we approximate \delta (g) over each spike by the average of its two extreme values, (\delta ({g}_{0})+{\delta}_{i})/2. This approximation yields

{\tilde{\Omega}}_{1}\approx \frac{{G}_{i}-({g}_{0}-\delta ({g}_{0})/2)}{(\delta ({g}_{0})+{\delta}_{i})/2}.

(16)

Similarly, once the input is used up, spikes continue to be fired as *g* decays from {g}_{0} to approximately (1-I)/(E-1), and the number of additional spikes that result is estimated by

{\tilde{\Omega}}_{2}\approx \frac{{g}_{0}+\delta ({g}_{0})/2-(1-I)/(E-1)}{(\delta ({g}_{0})+{\delta}_{f})/2}.

(17)

In the above equations, we have taken into account that the trajectory may not be reset precisely at {g}_{0} but rather somewhere within an interval approximated by ({g}_{0}-\delta ({g}_{0})/2,{g}_{0}+\delta ({g}_{0})/2). Because this may lead to an overestimation by one or two spikes, we set {\Omega}_{1}+{\Omega}_{2}={\tilde{\Omega}}_{1}+{\tilde{\Omega}}_{2}-1. The total number of spikes fired is finally estimated by

\Omega ={\Omega}_{1}+{\Omega}_{2}+{\Omega}_{3}

(18)

using equations (13)-(17).

The calculation can be easily generalized for input patterns that push *g* above and below {g}_{0} multiple times, although they will be non-optimal by our earlier arguments. Similarly, for an initial kick {G}_{i}<{g}_{0} and g<{g}_{0} for all time, the smallest \delta (g) available is \delta ({G}_{i}) and we can estimate the number of spikes resulting from partitioning the input into kicks of size \delta ({G}_{i}) by the equation

\Omega =\frac{G-{G}_{i}}{{\delta}_{i}}+\frac{{G}_{i}-(1-I)/(E-1)}{({\delta}_{i}+{\delta}_{f})/2}.

(19)

If E+I-2EI\ge 0, then the same calculations still apply. The big kick strategy is optimal here, yielding a number of spikes estimated by

\Omega =\frac{G-(1-I)/(E-1)}{(\delta (G)+{\delta}_{f})/2}.

(20)

Generalizing, a strategy of giving an initial input {G}_{i}, followed by repeated kicks of size {\delta}_{i} until the input is depleted, yields a number of spikes estimated by

\Omega =\frac{G-{G}_{i}}{{\delta}_{i}}+\frac{{G}_{i}-(1-I)/(E-1)}{({\delta}_{i}+{\delta}_{f})/2}.

(21)

Figure 4 shows comparisons of our spike estimates from equations (18) and (21) with numerical computed counts of spikes, illustrating that our estimates can be reasonable.

In fact, in the case of E+I-2EI\ge 0, we underestimate the number of spikes fired for large *G* and {G}_{i}. This underestimation results because we average \delta (G) or {\delta}_{i} with {\delta}_{f} in the denominator of equation (20) or (21), whereas most spikes yield decreases in *g* that are much smaller than {\delta}_{f}. Improved estimates in such cases be obtained by weighting this denominator more toward \delta (G) or {\delta}_{i}, which will decrease the denominator and thus will always yield predictions of additional spikes for larger kicks, relative to the formulas in equations (20), (21). An example resulting from extreme weighting, replacing the average of {\delta}_{i} and {\delta}_{f} with {\delta}_{i} alone, is also shown in Figure 4, as is a similar example for the case of E+I-2EI<0.

In summary, equations of the form (18)-(21), each requiring calculation of only a small number of quantities, can be used on a case by case basis to estimate the numbers of spikes that will result from a given strategy and therefore to compare strategies. These formulas provide for an informed comparison between the two types of big kick strategies determined to be optimal for the two distinct cases of E+I-2EI\ge 0 and E+I-2EI<0, respectively, and the critical kicks strategy. Furthermore, now that we have defined \delta (g), we can give a more precise variation on the calculation of equation (9) made in the subsection on phase plane structures and basic strategies to show that truly optimal strategies (other than the critical kick strategy based on {\tilde{\gamma}}_{c}) provide kicks with v=I, so each strategy should include a time shift so that kicks are given when this condition is met, rather than with v=0 or v=1. Specifically, if {g}_{1}>{g}_{2}, then for {v}_{0}\in [0,1],

\begin{array}{rl}\delta ({g}_{1},{v}_{0})-\delta ({g}_{2},{v}_{0})=& {\int}_{0}^{{v}_{0}}\frac{\beta {g}_{1}}{I-v-{g}_{1}(v-E)}\phantom{\rule{0.2em}{0ex}}dv\\ -{\int}_{0}^{{v}_{0}}\frac{\beta {g}_{2}}{I-v-{g}_{2}(v-E)}\phantom{\rule{0.2em}{0ex}}dv.\end{array}

Calculating the derivative of the above equation with respect to {v}_{0} yields

\frac{d(\delta ({g}_{1},{v}_{0})-\delta ({g}_{2},{v}_{0}))}{d{v}_{0}}=\frac{\beta (I-{v}_{0})({g}_{1}-{g}_{2})}{(I-{v}_{0}-{g}_{1}{v}_{0}+{g}_{1}E)(I-{v}_{0}-{g}_{2}{v}_{0}+{g}_{2}E)},

a quantity that is positive for {v}_{0}<I and negative for {v}_{0}>I. Hence, the additional input needed to cross bands is minimal for kicks given at {v}_{0}=I, in agreement with Proposition 1. Finally, it is not difficult to see from examination of the above spike counts and equation (10) that, with other parameters fixed, increases in *β* yield fewer spikes, as expected from the corresponding faster decay of *g*, while increases in *E* and *I* yield more spikes, as expected from the increased rate of change of *v*.