Abstract

Ramping neuronal activity refers to spiking activity with a rate that increases quasi-linearly over time. It has been observed in multiple cortical areas and is correlated with evidence accumulation processes or timing. In this work, we investigated the downstream effect of ramping neuronal activity through synapses that display short-term facilitation (STF) or depression (STD). We obtained an analytical result for a synapse driven by deterministic linear ramping input that exhibits pure STF or STD and numerically investigated the general case when a synapse displays both STF and STD. We show that the analytical deterministic solution gives an accurate description of the averaging synaptic activation of many inputs converging onto a postsynaptic neuron, even when fluctuations in the ramping input are strong. Activation of a synapse with STF shows an initial cubical increase with time, followed by a linear ramping similar to a synapse without STF. Activation of a synapse with STD grows in time to a maximum before falling and reaching a plateau, and this steady state is independent of the slope of the ramping input. For a synapse displaying both STF and STD, an increase in the depression time constant from a value much smaller than the facilitation time constant to a value much larger than leads to a transition from facilitation dominance to depression dominance. Therefore, our work provides insights into the impact of ramping neuronal activity on downstream neurons through synapses that display short-term plasticity. In a perceptual decision-making process, ramping activity has been observed in the parietal and prefrontal cortices, with a slope that decreases with task difficulty. Our work predicts that neurons downstream from such a decision circuit could instead display a firing plateau independent of the task difficulty, provided that the synaptic connection is endowed with short-term depression.

1  Introduction

Ramping neuronal activity has been observed in different cortical and subcortical areas, such as lateral intraparietal cortex, frontal eye field, superior colliculus, thalamus, and presupplementary and supplementary motor areas, and it provides a neuronal implementation of the evidence accumulation process during perceptual decision making (Wang 2002, 2008; Gold & Shadlen, 2007) and timing (Komura et al., 2001; Reutimann, Yakovlev, Fusi, & Senn, 2004; Mita, Mushiake, Shima, Matsuzaka, & Tanji, 2009; Simen, Balci, de Souza, Cohen, & Holmes, 2011; Merchant, Harrington, & Meck, 2013). A simple mathematical model for the ramping neuronal activities is the drift-diffusion model (Hanes & Schall, 1996; Huk & Shadlen, 2005; Ratcliff, Cherian, & Segraves, 2003). Neurons in the caudate nucleus (the eye movement part of the striatum) have been observed to show a ramping activity followed by a saturation before saccade initiation (Ding & Gold, 2010). The striatum receives direct projections from cortical areas that display ramping neuronal activity. The origin of the saturation of ramping activity in the striatum is not yet well understood.

Short-term plasticity (STP) is a common feature of cortical synapses, and both short-term facilitation (STF) and depression (STD) have been observed in the cortex (Abbott & Regehr, 2004; Morrison, Diesmann, & Gerstner, 2008). In the phenomenological model of STP, depression is attributed to a decrease of vesicle availability, while facilitation is attributed to an increase of vesicle release probability (Tsodyks & Markram, 1997; Tsodyks, Pawelzik, & Markram, 1998; Fuhrmann, Segev, Markram, & Tsodyks, 2002; Hempel, Hartman, Wang, Turrigiano, & Nelson, 2000). For simplicity in theoretical understanding of the role of synaptic plasticity, depression or facilitation alone was often considered, although in general, both depression and facilitation coexist in a synapse. In experimental and theoretical investigations, presynaptic input with a constant rate is usually applied. In previous work, we proposed that the saturation of striatal ramping during evidence accumulation could be explained by STD in the corticostriatal synapses (Wei, Rubin, & Wang, 2015). In this work, we analytically characterized the downstream effect of synapses displaying STF or STD driven by fluctuating ramping presynaptic inputs described by the drift-diffusion model. We also investigated the general case when both STF and STD coexist in a synapse numerically.

2  Phenomenological Model for STP

In the phenomenological model for STP, the activity of a synapse with pure STF is described as
formula
2.1
formula
2.2
where tj is the time of the jth presynaptic input spike, F represents the vesicle release probability, and s is the gating variable. When a presynaptic spike arrives, F first increases with an amount , and then decays with a time constant during the intervals of input spikes. The gating variable s is then updated following the update of F, with an increment of F. The conductance to a postsynaptic neuron is given by the product of the gating variable s and a constant synaptic efficacy.
The presynaptic spike train in equations 2.1 and 2.2 represents the instantaneous input firing rate , that is, . Here we are interested in the case when describes a ramping input rate as observed for cortical neurons in multiple brain areas. The ramping input rate is well described by the classical drift-diffusion model,
formula
2.3
where is the time constant for the drift-diffusion process, is the drift term, is a gaussian white noise satisfying and , and is the noise strength. We will take , but all the results will not depend on the exact value of . The synaptic dynamics are given by the following equations,
formula
2.4
formula
2.5
We will use , , and . The value for is characteristic of the decay time constant for AMPAergic synapses. Different values of and (with ) will not qualitatively change the behavior of the phenomenological model.
For a synapse with pure STD, the synaptic dynamics are described as
formula
2.6
formula
2.7
where p is a constant vesicle release probability and D represents the fraction of available vesicles. At the arrival of a presynaptic spike, D is first reduced by an amount and then recovered with a time constant during the intervals of input spikes. The gating variable s is then updated following the update of D, with an increment of .
Replacing the presynaptic spike train by the instantaneous input rate described by equation 2.3, we have the following equations:
formula
2.8
formula
2.9
We will use and . Different values of p and (with ) will not qualitatively change the behavior of the phenomenological model.

We first study a deterministic linear ramping input ( case) and then consider a general fluctuating ramping input described by the drift-diffusion model . The initial values of F, D, and s at are denoted as F0, D0, and s0. We will use , , and unless stated otherwise.

3  Deterministic Linear Ramping Input

3.1  Synapse with STF

When in equation 2.3, the input rate ramps up linearly with time (i.e., ), where is the slope of ramping. Equations 2.4 and 2.5 can be solved analytically,
formula
3.1
formula
3.2
where is the Dawson’s integral (Abramowitz & Stegun, 1970) as defined in the appendix.
In Figure 1, the synaptic dynamics for ramping input with two different slopes and constant input (see Figure 1A) are illustrated and are compared with the control case when there is no STP (see Figure 1B). The activities of F and s for a synapse with STF are shown in Figures 1C and 1D. When t is small, we have
formula
3.3
formula
3.4
Figure 1:

Activation level of a synapse with pure STF or STD. (A) The input rate ramps up linearly, with slope (green) and 50 (blue) (in ). The black curve represents a constant input with rate 30 Hz. (B) The activation level of the gating variable s in the control case when there is no STP. (C,D) The facilitation factor F (C) and the activation level s (D) for a synapse with pure STF. (E,F) The depression factor D (E) and activation level s (F) for a synapse with pure STD. The gray shaded areas (with width ) highlight the different activity of s for a synapse with and without STP. Parameters used: ms, ms, , ms, .

Figure 1:

Activation level of a synapse with pure STF or STD. (A) The input rate ramps up linearly, with slope (green) and 50 (blue) (in ). The black curve represents a constant input with rate 30 Hz. (B) The activation level of the gating variable s in the control case when there is no STP. (C,D) The facilitation factor F (C) and the activation level s (D) for a synapse with pure STF. (E,F) The depression factor D (E) and activation level s (F) for a synapse with pure STD. The gray shaded areas (with width ) highlight the different activity of s for a synapse with and without STP. Parameters used: ms, ms, , ms, .

When t is large, we have
formula
3.5
formula
3.6
Therefore, when t is large, the facilitation factor F will saturate hyperbolically for a ramping input, while for a constant input, it will reach its stationary state exponentially (see Figure 1C and the appendix). The gating variable s will increase cubically at the initial period (gray region in Figure 1D, with width ), then is followed by a linear increase similar to the control case (see Figure 1B). Note that for a constant input rate, s will saturate when t is large in the timescale for a synapse with STF (see Figure 1D, black curve), while in the control case, s will saturate to its steady state in the much shorter timescale (see Figure 1B, black curve).

3.2  Synapse with STD

For and , from equations 2.8 and 2.9, we have
formula
3.7
formula
3.8
When t is large,
formula
3.9
formula
3.10
We see that for the ramping input, D decays hyperbolically for large t, which is slower than that for a constant input rate that decays exponentially to its steady state (see Figure 1E). The gating variable s will reach a steady state, which is independent of the ramping slope k and release probability p (see Figure 1F). Note that the steady state for a ramping input is the same as that for a constant input when the constant rate is large enough, as given in the appendix.
Before reaching its steady state, the gating variable s for a synapse with STD will develop a maximum when receiving ramping input (see Figure 1F). Although the steady state of s is independent of the ramping slope k and the release probability p, the maximal value of s depends on both k and p. For small t and applying equation A.3,
formula
3.11
where () is small, so we have
formula
3.12
By comparing equation 3.12 with equation A.9 in the appendix, we see the different small time behaviors of D for ramping and constant input rates (see Figure 1 E). Therefore, when t is small (but ),
formula
3.13
which has a maximal value at time given by
formula
3.14
formula
3.15
Figures 2A and 2B compare the approximate and real values of and as a function of k, respectively. We see that equations 3.14 and 3.15 provide qualitative information about the location and value of the maximum of s: a ramping input with a higher slope k will have a larger maximum developed within a shorter time.
Figure 2:

The maximal activation level and the time for reaching the maximum for a synapse with only STD as a function of ramping slope. (A) The time for reaching the maximum of s as a function of the ramping slope k. (B) The maximal value of s as a function of the ramping slope k. The dashed lines in panels A and B are approximations from equations 3.14 and 3.15, respectively.

Figure 2:

The maximal activation level and the time for reaching the maximum for a synapse with only STD as a function of ramping slope. (A) The time for reaching the maximum of s as a function of the ramping slope k. (B) The maximal value of s as a function of the ramping slope k. The dashed lines in panels A and B are approximations from equations 3.14 and 3.15, respectively.

3.3  Synapse with Both STF and STD

In general, both STF and STD are coexisting in a synapse, with the dynamics described by Tsodyks et al. (1998), Hempel et al. (2000), and Lindner, Gangloff, Longtin, and Lewis (2009):
formula
3.16
formula
3.17
formula
3.18
For a deterministic linear input rate, , the solution of equation 3.16 is given by equation 3.1. Since is much smaller than and , equation 3.18 is simplified to
formula
3.19
Equation 3.17, however, cannot be solved analytically. For large t, (see equation 3.5). When , the synapse is depression dominated, and (see equation 3.9, replacing p with ). Therefore when , we have
formula
3.20
the same long time limit as that for a synapse with pure STD.

We solve equation 3.17 numerically and show the time evolution of facilitation factor F, depression factor D, and gating variable s in Figure 3 for different model parameters. The top row of Figure 3 shows the dependence of synaptic dynamics on . Note that here, we illustrate a depression-dominated synapse. We see that with a larger , the increase of F and the decrease of D with time become faster, and s has a larger maximum and reaches its maximum faster (see panels A1–A3 in Figure 3). Modulating does not qualitatively change the synaptic dynamics (the same conclusion still holds for a facilitation-dominated synapse). We also see that the long time limit of s does not depend on (see panel A3 in Figure 3), as equation 3.20 suggests.

Figure 3:

Activation level of a synapse with both STF and STD. (A1–A3) The time evolution of facilitating factor F (A1), depressing factor D (A2), and gating variable s (A3) for different (green, 0.4; blue, 0.25; black, 0.1). (B1–B3) The time evolution of F (B1), D (B2), and s (B3) for different (green, 600 ms; blue, 300 ms; black, 100 ms). (C1–C3) The time evolution of F (C1), D (C2), and s (C3) for different (green, 600 ms; blue, 300 ms; black, 100 ms). Parameters used: , ms. In A1 to A3, ms, ms. In panels B1 to B3, ms, . In panels C1 to C3, ms, .

Figure 3:

Activation level of a synapse with both STF and STD. (A1–A3) The time evolution of facilitating factor F (A1), depressing factor D (A2), and gating variable s (A3) for different (green, 0.4; blue, 0.25; black, 0.1). (B1–B3) The time evolution of F (B1), D (B2), and s (B3) for different (green, 600 ms; blue, 300 ms; black, 100 ms). (C1–C3) The time evolution of F (C1), D (C2), and s (C3) for different (green, 600 ms; blue, 300 ms; black, 100 ms). Parameters used: , ms. In A1 to A3, ms, ms. In panels B1 to B3, ms, . In panels C1 to C3, ms, .

The dependence of synaptic dynamics on the facilitation time constant for a facilitation-dominated synapse is shown in the middle row of Figure 3. With a larger , the increase of F and s and the decrease of D become faster (see panels B1–B3 in Figure 3). When , D and s become insensitive to the value of (see the blue and green curves in panels B2 and B3, Figure 3). We see that after a period of rapid increase, the activation level of the gating variable s for a facilitation-dominated synapse increases much more slowly than that for a synapse with pure STF (compare the blue curves in Figure 1 D and in panel B3, Figure 3). For a depression-dominated synapse, D and s depend only weakly on (data not shown). The bottom row in Figure 3 shows the dependence of synaptic dynamics on the depression time constant . We see that the facilitation factor F is independent of (see panel C1 in Figure 3) by design (see equation 3.16), while the depression factor D decreases faster with a larger (see panel C2 in Figure 3). Interestingly, the gating variable s shows a transition from facilitation dominance to depression dominance when increases from a value much smaller than to a value much larger than (see the black and green curves in panel C3, Figure 3). Therefore, the interaction of F and D determines whether the synapse is facilitation or depression dominated.

4  Fluctuating Ramping Input

We now consider the case when the ramping input is noisy and is described by equation 2.3, as neurons downstream from a decision circuit will receive. Each downstream neuron receives a summation of inputs from many presynaptic neurons. Since the synaptic efficacy scales with the number of presynaptic neurons, the conductance to a downstream neuron is given by the average gating variables, denoted as . Here, the angular bracket represents averaging over presynaptic neuronal populations, which is implemented in simulations by averaging different realizations of the noise. The upper row of Figure 4 shows 10 different realizations of (see panel A), gating variable s for synapses with pure STF (see panel B) and with pure STD (see panel C), and the corresponding averaging values over 300 realizations for (i.e., sec−2).

Figure 4:

Synaptic activation of a downstream neuron that receives inputs from a population of presynaptic neurons showing ramping activity. (A–C) Fluctuating ramping input described by the drift-diffusion model (A), the activation level s for synapses with pure STF (B), and for synapses with pure STD (C). Blue: the average activity of (A) and s (B–C) over 300 realizations. Green: illustrated trajectories for 10 realizations. In panels A to C, () and . (D–F) Ensemble averaged ramping inputs (D), activation level for synapses with pure STF (E), and pure STD (F) for different noise levels with , (in sec−1). N is the size of presynaptic population.

Figure 4:

Synaptic activation of a downstream neuron that receives inputs from a population of presynaptic neurons showing ramping activity. (A–C) Fluctuating ramping input described by the drift-diffusion model (A), the activation level s for synapses with pure STF (B), and for synapses with pure STD (C). Blue: the average activity of (A) and s (B–C) over 300 realizations. Green: illustrated trajectories for 10 realizations. In panels A to C, () and . (D–F) Ensemble averaged ramping inputs (D), activation level for synapses with pure STF (E), and pure STD (F) for different noise levels with , (in sec−1). N is the size of presynaptic population.

Figure 4 (lower row) shows the results for Hz when Hz. As one expects, the average ramping input follows the deterministic trajectory (see Figure 4D). Since equations 2.4 and 2.8 cannot be solved analytically due to the interaction term between and F or D, we hope the deterministic results obtained in the preceding section can give a good approximation to the average synaptic activation in the low-noise regime, that is, when is small. From Figures 4E and 4F, we see that the deterministic results give a very accurate approximation ( in Figure 4E for STF, in Figure 4F, when sec−1, respectively), even when the noise level is two times the drift term. Therefore, the theoretical results for deterministic ramping input provide a good description for the conductance of downstream neurons receiving ramping inputs described by the drift-diffusion model.

5  Discussion

We investigated the downstream effect of ramping neuronal activity through synapses showing STP. We first derived analytical results for deterministic linear ramping input through synapses displaying pure STF or STD and showed the different synaptic dynamics in contrast to a steady input rate. We obtained a simple approximation for the maximal value of the activation level and the time taken to reach the maximum for a synapse with pure STD. For the general case, when a synapse shows both STF and STD effects, we found that it behaves as a synapse with pure STD when the facilitation time constant is much smaller than the depression time constant . We further investigated the dependence of synaptic dynamics on model parameters numerically in the general case. We found a transition from facilitation dominance to depression dominance when increases from a value much smaller than to a value much larger than . A downstream neuron will receive an averaged input from the presynaptic population. We then showed that the averaged synaptic activation level to the downstream neuron is well described by the deterministic solution even when the fluctuation of the ramping input is strong. These results provide insights for downstream impacts of ramping neuronal activity through synapses showing STP.

Cortical areas showing ramping neuronal activity project to other cortical areas and also subcortical areas such as the striatum, thalamus, and superior colliculus. The striatal activity has been observed to ramp up followed by saturation during evidence accumulation when monkeys performed perceptual decision making (Ding & Gold, 2010). Our work showed that this observation could be explained by utilizing STD in the corticostriatal synaptic projection. A direct measurement of EPSP/EPSC for synapses receiving ramping input is not available yet. Our results could be directly tested by intracellular recording in vitro (e.g., using cortical slices), in which a synaptic input ramps over time with a slope that can be parametrically varied. For synapses with STF, our results showed that the activation of gating variable is slower and less sensitive to the ramping slope at the early stage than the control case without STP, followed by the same linear increase as in the control case since for large t. For synapses with STD, the ramping slope is encoded only at the initial period of evidence accumulation and the steady state is insensitive to it. This prediction can be tested in the fixation time version of perceptual decision-making tasks. Therefore, we showed the differential properties of synapses with STF and STD in encoding the slope of ramping activity, which represents the bias of stimulus in making decisions.

One future direction is to include the intrinsic synaptic noise in the deterministic model for STP and investigate its influence when the synapse is driven by fluctuating ramping input. Depressing synapses with intrinsic noise receiving a Poisson input with constant rate have been studied (Rosenbaum, Rubin, & Doiron, 2012, 2013). Synapses displaying STD and stochastic vesicle dynamics were shown to behave as a frequency-dependent filter in signal transmission (Matveev & Wang, 2000; Rosenbaum et al., 2012), in contrast to the broadband signal transmission for deterministic synapses (Lindner et al., 2009), and also influence the transfer of neuronal correlations (Rosenbaum et al., 2013). Extension of our work to include intrinsic synaptic noise will provide further insights into the downstream impact of ramping neuronal activity during perceptual decision making and timing.

Appendix 

The Dawson’s integral is defined as (Abramowitz & Stegun, 1970)
formula
A.1
has the following asymptotic expansion
formula
A.2
when x is large, and the series expansion
formula
A.3
when x is small (Abramowitz & Stegun, 1970).
For comparison, we present here results for a constant input rate, . For a synapse with pure STF, the solutions for equations 2.4 and 2.5 are given by
formula
A.4
formula
A.5
For large t,
formula
A.6
For a synapse with pure STD, the solutions of equations 2.8 and 2.9 are given by
formula
A.7
formula
A.8
When t is small (but larger than ),
formula
A.9
When t is large, D and s reach their steady states,
formula
A.10
formula
A.11
When , that is, Hz, this steady state of s is reduced to
formula
A.12
which is independent of the constant input rate r and release probability p and has the same expression as the steady state for a deterministic linear ramping input, equation 3.10.
As a control case, we consider a synapse without STP. Taking in equation 2.5, then the activation of the synapse is given by
formula
A.13
for a ramping input rate and
formula
A.14
for a constant input rate r.

Acknowledgments

This work was supported by a Swartz Foundation Fellowship (W.W.), and the National Institutes of Health grants R01 MH062349 (X.-J. W).

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