Abstract
Temporal filters, the ability of postsynaptic neurons to preferentially select certain presynaptic input patterns over others, have been shown to be associated with the notion of information filtering and coding of sensory inputs. Shortterm plasticity (depression and facilitation; STP) has been proposed to be an important player in the generation of temporal filters. We carry out a systematic modeling, analysis and computational study to understand how characteristic postsynaptic (low, high and bandpass) temporal filters are generated in response to periodic presynaptic spike trains in the presence STP. We investigate how the dynamic properties of these filters depend on the interplay of a hierarchy of processes, including the arrival of the presynaptic spikes, the activation of STP, its effect on the excitatory synaptic connection efficacy, and the response of the postsynaptic cell. These mechanisms involve the interplay of a collection of time scales that operate at the singleevent level (roughly, during each presynaptic interspikeinterval) and control the longterm development of the temporal filters over multiple presynaptic events. These time scales are generated at the levels of the presynaptic cell (captured by the presynaptic interspikeintervals), shortterm depression and facilitation, synaptic dynamics and the postsynaptic cellular currents. We develop mathematical tools to link the singleevent time scales with the time scales governing the longterm dynamics of the resulting temporal filters for a relatively simple model where depression and facilitation interact at the level of the synaptic efficacy change. We extend our results and tools to account for more complex models. These include multiple STP time scales and nonperiodic presynaptic inputs. The results and ideas we develop have implications for the understanding of the generation of temporal filters in complex networks for which the simple feedforward network we investigate here is a building block.
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Acknowledgements
This work was partially supported by the National Science Foundation grant DMS1608077 (HGR) and an NSF Graduate Research Fellowship (YM). The authors are grateful to Allen Tannenbaum for useful comments and support, and to Farzan Nadim, Dirk Bucher and Nelly Daur for useful discussions.
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Appendices
A. 1D linear difference equations
1.1 A.1 Constant coefficients
Consider the following linear difference equation
where \(\alpha\) and \(\beta\) are constants. The steadystate for this equation, if it exists, is given by
By solving (73) recurrently and using
where \(a \ne 1\) is a real number, one gets
Substitution of (74) into this equation yields
Application of formula (77) to the difference Eqs. (7) and (8) gives, respectively,
and
1.2 A.2 Variable (ndependent) coefficients
Consider the following linear difference equation
By solving (73) recurrently one gets
where we are using the convention \(\prod _{j_1}^{j_2} = 1\) if \(j_1 > j_2\). Equation (81) reduces to Eq. (79) if both coefficients in (81) are constant.
Consider now Eq. (80) where the coefficients are expressed as small perturbations \(\delta _{\alpha ,n} \ll 1\) and \(\delta _{\beta ,n} \ll 1\) (\(n = 1, 2, \ldots\)), respectively, of constant coefficients
To the first order approximation, the solution (81) reads
B. Some properties of \(\bar{X}\) and \(\bar{Z}\) and their dependence with \(\Delta _{spk}\) and \(\tau _{dep/fac}\)
Consider \(\bar{X}\) and \(\bar{Z}\) given by (9) and (10), respectively.
1.1 B.1 Monotonic dependence of \(\bar{X}\) and \(\bar{Z}\) with \(\Delta _{spk}\)
If \(a_d > 0\) and \(x_{\infty } > 0\), then \(\bar{X}\) is an increasing function of \(\Delta _{spk}\) and a decreasing function of \(f_{spk}\). This results from
If \(a_f < 1\) and \(z_{\infty } < 1\), then \(\bar{Z}\) is a decreasing function of \(\Delta _{spk}\) and an increasing function of \(f_{spk}\). This results from
1.2 B.2 Monotonic dependence of \(\bar{X}\) and \(\bar{Z}\) with \(\tau _{dep/fac}\)
If \(a_d > 0\) and \(x_{\infty } > 0\), then \(\bar{X}\) is a decreasing function of \(\tau _{dep}\). This results from
If \(a_f < 1\) and \(z_{\infty } < 1\), then \(\bar{Z}\) is a decreasing function of \(\tau _{fac}\). This results from
C. Models of synaptic depression and facilitation
1.1 C.1 Depression  facilitation model used in Latorre et al. (2016)
Following Destexhe et al. (1994, 1998), the synaptic variables S obey a kinetic equation of the form
where N(V) (mM) representes the neurotransmitter concentration in the synaptic cleft. Neurotransmitters are assumed to be released quickly upon the arrival of a presynaptic spike and remain in the synaptic cleft for the duration of the spike (\(\sim 1\) ms). This can be modeled by either using a sigmoid function
or a step function if the release is assumed to be instantaneous. The parameters \(\tau _r\) and \(\tau _d\) are the rise and decay time constants respectively (msec).
This model assumes N(V) is independent of the spiking history (the value of N(V) during a spike is constant, except possibly for the dependence on V). (There is evidence that this is not realistic Markram and Tsodyks (1996), Dudel and Kuffler (1961).) In Latorre et al. (2016), the “activated” time was 1 ms Clements et al. (1992), Colquhoun et al. (1992).
In Latorre et al. (2016), they followed the description of the synaptic shortterm dynamics following Tsodyks and Markram (1997), Tsodyks et al. (1998) (Sect. 2.1.5). For the dynamics of the synaptic function S, they used a function \([T] = \kappa \, \Delta S_n\) during the release time and \([T] = 0\) otherwise, instead of N(V). The combination of the two formulations yields
In the following alternative formulation Drover et al. (2007) \(\kappa \, \Delta S_n\) does not affect the effective rise time of the synaptic function S
1.2 C.2 Depression model used in Manor and Nadim (2001)
Following experimental procedures described in Manor et al. (1997), the synaptic current is described by \(I_{syn}=G_{ex} a\, d (VE_{ex})\) where a and d are variables that represent activation and depression processes, respectively. They follow the form:
where \(y=a,d\). The steadystate of y is given by
and its time constant follows
This model is used in Manor and Nadim (2001) to describe bistability in pacemaker networks with recurrent inhibition and depressing synapses. Parameters in these equations are experimentally fitted from the pyloric network of the crab Cancer borealis.
D. Additional model formulations for multiple depressionfacilitation processes
In Sect. 3.7 we discussed the model formulation (47) and (48) describing the interplay of two depressionfacilitations processes. A number of additional, simplified formulations are possible based on different assumptions. The models we propose here are natural mathematical extensions of the single depression/facilitation processes discussed in the main body of this paper. They are phenomenological models, not based on any experimental observation or theoretical foundation, and they are limited in their general applicability. However, they are useful to explore the possible scenarios underlying the interplay of multiple depression and facilitation time scales affecting the PSP dynamics of a cell in response to presynaptic input trains.
1.1 D.1 Additive and multiplicative segregatedprocesses models
In the additive and multiplicative segregated models, the variable M is given, respectively, by
and
where the parameter \(\alpha \in [0,1]\) controls the relative contribution of each of the processes. Correspondingly, the updates are given by
and
For \(\alpha = 0\), \(\Delta S_n^+\) and \(\Delta S_n^*\) reduce to \(\Delta S_{1,n}\) (single depressionfacilitation process). This accounts for the regimes where \(\tau _{dep,2}, \tau _{fac,2} \ll 1\). If the two processes are equal (\(\tau _{dep,1} = \tau _{dep,2}\) and \(\tau _{fac,1} = \tau _{fac,2}\)), then \(\Delta S_n^+\) and \(\Delta S_n^*\) also reduce to \(\Delta S_{1,n}\). However, these models fail to account for the reducibility in the situations where only \(\tau _{dep,2} \ll 1\) or \(\tau _{fac,2} \ll 1\), but not both. The option of considering depression to be described by \(x_1\) and facilitation by \(z_ 2\) (with \(\tau _{fac,1}, \tau _{dep,2} \ll 1\)) is technically possible in the context of the model, but it wouldn’t be consistent with the model description of single depressionfacilitation processes, and it will make no sense to use the model in this way. In general, this model would be useful when the depression and facilitation time scales for each process 1 and 2 are comparable and the differences in these time scales across depression/facilitation processes should be large enough.
1.2 D.2 Fully multiplicative model
One natural way to extend the variable M to more than one process is by considering
and the synaptic update, given by
This formulation presents us with a number of consistency problems related to the reducibility (or lack of thereoff) to a single depressionfacilitation process in some limiting cases when, for example, the two depression or facilitation time constants are very similar and therefore the associated processes are almost identical, or the depression or facilitation time constants are very small and therefore the envelopes of the associated processes are almost constant across cycles.
More specifically, first, if \(\tau _{dep,2}, \tau _{fac,2} \ll 1\) (almost no STD), then \(X_{2,n} Z_{2,n} \sim \bar{X}_2 \bar{Z}_2 = a_f\) for all n after a very short transient and therefore \(\Delta S_n^\# = X_1 Z_1 a_f \ne \Delta S_{1,n}\). One way, perhaps the simplest, to address this is to divide the expressions (99 ) and (100 ) by \(a_f^2\) and redefine \(\Delta S_{k,n}\) for the single depressionfacilitation process accordingly. Specifically,
where we use the notation
The effect of redefining \(\Delta S_{k,n}\) by dividing the original expression (used in the previous sections) does not affect the time constants and the differences in the values between the two formulations is absorbed by the maximal synaptic conductance.
Second, if \(\tau _{dep,1} = \tau _{dep,2}\) and \(\tau _{fac,1} = \tau _{fac,2}\), then \(X_{1,n} = X_{2,n}\) and \(Z_{1,n} = Z_{2,n}\) for all n, and \(\Delta S_n^\# = \Delta S_{1,n}^2\) instead of \(\Delta S_n^\# = \Delta S_{1,n}\). In order to address this, the synaptic update can be modified to
where
and \(H(\Delta \tau )\) is a rapidly decreasing function satisfying \(H(0) = 2\) and \(\lim _{\Delta \tau \rightarrow \infty } H(\Delta \tau ) = 1\). In our simulations we will use
with \(\beta > 0\). Correspondingly,
In this way,

If \(\tau _{dep,1} = \tau _{dep,2}\), then \(X_{1,n} = X_{2,n}\) for all n and \(\lambda _{dep} = 1/2\). This gives
$$\begin{aligned} \Delta S_n^\# = X_{1,n} \left[ \frac{Z_{1,n} Z_{2,n}}{a_f^2} \right] ^{\lambda _{fac}} . \end{aligned}$$If, in addition, \(\tau _{fac,1} \ne \tau _{fac,2}\) and \( \tau _{fac,1}\tau _{fac,2} > 0\) is large enough, then \(\lambda _{fac} = 1\) and
$$\begin{aligned} \Delta S_n^\# = X_{1,n} \frac{Z_{1,n} Z_{2,n}}{a_f^2} = \Delta S_{1,n} \frac{Z_{2,n}}{a_f}. \end{aligned}$$ 
If \(\tau _{fac,1} = \tau _{fac,2}\), then \(Z_{1,n} = Z_{2,n}\) for all n, \(\lambda _{fac} = 2\) and
$$\begin{aligned} \Delta S_n^\# = \left[ X_{1,n} X_{2,n} \right] ^{\lambda _{dep}} \frac{Z_{1,n}}{a_f}. \end{aligned}$$If, in addition, \(\tau _{dep,1} \ne \tau _{dep,2}\) and \( \tau _{dep,1}\tau _{dep,2} > 0\) is large enough, then \(\lambda _{dep} = 1\) and
$$\begin{aligned} \Delta S_n^\# = X_{1,n} X_{2,n} \frac{Z_{1,n}}{a_f} = \Delta S_{1,n} X_{2,n}. \end{aligned}$$ 
It follows that if both \(\tau _{dep,1} = \tau _{dep,2}\) and \(\tau _{fac,1} = \tau _{fac,2}\), then \(X_{1,n} = X_{2,n}\) and \(Z_{1,n} = Z_{2,n}\) for all n, \(\lambda _{dep} = \lambda _{fac} = 2\) and
$$\begin{aligned} \Delta S_n^\# = X_{1,n} \frac{Z_{1,n}}{a_f} = \Delta S_{1,n}. \end{aligned}$$ 
If \(\tau _{dep,2} \ll 1\) and \( \tau _{dep,1}\tau _{dep,2}\) is large enough, then \(X_{2,n} = 1\) for all n (after a very short transient), \(\lambda _{dep} = 1\), and then
$$\begin{aligned} \Delta S_n^\# = X_{1,n} \left[ \frac{Z_{1,n} Z_{2,n}}{a_f^2} \right] ^{\lambda _{fac}}. \end{aligned}$$If, in addition, \(\tau _{dep,1} \ll 1\) and \(\tau _{dep,2} \sim \tau _{dep,1}\) ( \( \tau _{dep,1}\tau _{dep,2} \sim 0\) not large enough), then \(X_{1,n} = 1\) for all n (after a very short transient), \(\lambda _{dep} = 2\), and then
$$\begin{aligned} \Delta S_n^\# = \left[ \frac{Z_{1,n} Z_{2,n}}{a_f^2} \right] ^{\lambda _{fac}}. \end{aligned}$$ 
If \(\tau _{fac,2} \ll 1\) and \( \tau _{fac,1}\tau _{fac,2}\) is large enough, then \(Z_{2,n} = a_f\) for all n (after a very short transient), \(\lambda _{fac} = 1\), and then
$$\begin{aligned} \Delta S_n^\# = \left[ X_{1,n} X_{2,n} \right] ^{\lambda _{dep}} \frac{Z_{1,n}}{a_f}. \end{aligned}$$If, in addition, \(\tau _{fac,1} \ll 1\) and \(\tau _{fac,2} \sim \tau _{face,1}\) ( \( \tau _{fac,1}\tau _{fac,2} \sim 0\) not large enough), then \(Z_{1,n} = a_f\) for all n (after a very short transient), \(\lambda _{fac} = 2\), and then
$$\begin{aligned} \Delta S_n^\# = \left[ X_{1,n} X_{2,n} \right] ^{\lambda _{dep}}. \end{aligned}$$ 
It follows that if \(\tau _{dep,1}, \tau _{dep,2} \ll 1\) ( \( \tau _{dep,1}\tau _{dep,2} \sim 0\) not large enough) and \(\tau _{fac,1}, \tau _{fac,2} \ll 1\) ( \( \tau _{fac,1}\tau _{fac,2} \sim 0\) not large enough), then
$$\begin{aligned} \Delta S_n^\# = 1. \end{aligned}$$
E. Descriptive rules for the generation of temporal (envelope) bandpass filters from the interplay of the temporal (envelope) low and highpass filters
From a geometric perspective, temporal bandpass filters in response to periodic presynaptic inputs arise as the result of the product of two exponentially increasing and decreasing functions both decaying towards their steadystate (e.g., Fig. 6). At the descriptive level, this is captured by the temporal envelope functions (F, G and \(H = F G\)) discussed above whose parameters are not the result of a sequence of single events but are related to the biophysical model parameters by comparison with the developed temporal filters. These functions provide a geometric/dynamic way to interpret the generation of temporal filters in terms of the properties of depression (decreasing functions) and facilitation (increasing functions) in response to periodic inputs, although this interpretation uses the developed temporal filters and therefore is devoid from any biophysical mechanistic interpretation.
In order to investigate how the multiplicative interaction between F(t) and G(t) given by Eqs. (31) and (32) give rise to the temporal bandpass filters \(H = F G\), we consider a rescaled version of these functions
and
where \(B = 1\) and
The function G transitions from \(G(0) = 1  C\) to \(\lim _{t \rightarrow \infty } G(t) = 1\) with a fixed time constant (Fig. 15, green curves). The function F transitions from \(F(0) = 1\) to \(\lim _{t \rightarrow \infty } F(t) = A\) with a time constant \(\eta\) (Fig. 15, red curves). It follows that H transitions from \(H(0) = 1  C\) to \(\lim _{t \rightarrow \infty } H(t) = A\, B = A\) (Fig. 15, blue curves). A temporal bandpass filter is generated if H raises above A for a range of values of t. This requires F to decay slow enough so within that range \(H = F G > A\) (Fig. 15A) or A to be small enough (Fig. 15B). In fact, as A decreases, the values of \(\eta\) required to produce a bandpass temporal filter increases (compare Fig. 15A2 and B2).
Changes in the parameter B in (108) affect the height of the bandpass temporal filter, but not the generation mechanism described above. However, for certain ranges of parameter values H is a temporal lowpass filter (not shown).
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Mondal, Y., Pena, R.F.O. & Rotstein, H.G. Temporal filters in response to presynaptic spike trains: interplay of cellular, synaptic and shortterm plasticity time scales. J Comput Neurosci 50, 395–429 (2022). https://doi.org/10.1007/s1082702200822y
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DOI: https://doi.org/10.1007/s1082702200822y