Synaptic filtering of rate-coded information

Matthias Merkel and Benjamin Lindner

Phys. Rev. E 81, 041921 – Published 29 April 2010

Abstract

In this paper, we analytically examine the influence of synaptic short-term plasticity (STP) on the transfer of rate-coded information through synapses. STP endows each presynaptic input spike with an amplitude that depends on previous input spikes. We develop a method to calculate the spectral statistics of this amplitude modulated spike train (postsynaptic input) for the case of an inhomogeneous Poisson process. We derive in particular analytical approximations for cross-spectra, power spectra, and for the coherence function between the postsynaptic input and the time-dependent rate modulation for a specific model. We give simple expressions for the coherence in the limiting cases of pure facilitation and pure depression. Using our analytical results and extensive numerical simulations, we study the spectral coherence function for postsynaptic input resulting from a single synapse or from a group of synapses. For a single synapse, we find that the synaptic coherence function is largely independent of frequency indicating broadband information transmission. This effect is even more pronounced for a large number of synapses. However, additional noise gives rise to frequency-dependent information filtering.

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Received 3 December 2009

DOI:https://doi.org/10.1103/PhysRevE.81.041921

Authors & Affiliations

Matthias Merkel and Benjamin Lindner

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany

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Vol. 81, Iss. 4 — April 2010

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Figure 1
Scheme illustrating Eqs. (1, 2, 3, 9): The external stimulus $R (t)$ modulates the rate of Poissonian spike trains $x_{0, i} (t)$ . Each of these spike trains is modulated by a dynamic synapse according to the FD model turning it into $x_{i} (t)$ . Finally, the total postsynaptic input $X (t)$ to the target neuron is given by the sum of these modulated spike trains with an additional noise $η (t)$ . The additional noise accounts for input from synapses that are not involved in the transmission of the stimulus signal $R$ .Reuse & Permissions
Figure 2
(Color online) Illustration of facilitation-only dynamics as described by Eq. (3) with $A (t) = F (t)$ , and Eqs. (6, 7). Upper panel: presynaptic spike train $x_{0} (t)$ . Middle panel: dynamics of $F_{C} (t)$ and facilitation dynamics $F (t)$ . Lower panel: postsynaptic input $x (t)$ . For illustration purposes, the presynaptic spike times were chosen manually. Parameters: $F_{0} = 0.1$ , $Δ = 0.3$ , and $τ_{F} = 200 ms$ .Reuse & Permissions
Figure 3
(Color online) Illustration of depression-only dynamics as described by Eq. (3) with $A (t) = F_{0} \cdot D (t)$ and Eq. (8) with $F (t) = F_{0}$ . Upper panel: presynaptic spike train $x_{0} (t)$ . Middle panel: depression dynamics $D (t)$ and synaptic amplitude $A (t)$ . Lower panel: postsynaptic input $x (t)$ . Parameters: $F_{0} = 0.4$ and $τ_{D} = 2000 ms$ ; spike times as in Fig. 2.Reuse & Permissions
Figure 4
(Color online) Illustration of the full FD dynamics as described by Eq. (3) with $A (t) = F (t) \cdot D (t)$ and Eqs. (6, 7, 8). Upper panel: presynaptic spike train $x_{0} (t)$ . Middle panel: facilitation dynamics $F (t)$ , depression dynamics $D (t)$ , and synaptic amplitude $A (t)$ . Lower panel: postsynaptic input $x (t)$ . Parameters: $F_{0} = 0.1$ , $Δ = 0.3$ , $τ_{F} = 500 ms$ , and $τ_{D} = 2000 ms$ ; spike times as in Fig. 2.Reuse & Permissions
Figure 5
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ for the case of facilitation only. The parameter values are $F_{0} = 0.1$ , $Δ = 0.3$ , $r = 10 Hz$ , $τ_{F} = 80 ms$ , and $ε = 0.2$ . Circles show the results of a simulation without any approximations made (error bars within symbol size); while the solid lines illustrate the theoretical predictions expressed in Eqs. (38, 41, 34), respectively. The spectra show pronounced low-pass behavior while the coherence function is rather flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ].Reuse & Permissions
Figure 6
(Color online) Theoretical results for ${| S_{R x} (0) |}^{2} / {| S_{R x} (\infty) |}^{2}$ and for $C_{R x} (0) / C_{R x} (\infty)$ . (A) ${| S_{R x} (0) |}^{2} / {| S_{R x} (\infty) |}^{2}$ depending on $r τ_{F}$ and the facilitation intensity expressed by the parameters for linear facilitation dynamics $Δ_{lin} / F_{0, lin}$ . (B) ${| S_{R x} (0) |}^{2} / {| S_{R x} (\infty) |}^{2}$ for $F_{0} = 0.04$ , depending on $r τ_{F}$ and $Δ$ . (C) $C_{R x} (0) / C_{R x} (\infty)$ depending on $r τ_{F}$ and $Δ_{lin} / F_{0, lin}$ . (D) $C_{R x} (0) / C_{R x} (\infty)$ for $F_{0} = 0.04$ , depending on $r τ_{F}$ and $Δ$ . The $Δ$ axes in (B) and (D) are scaled such that $Δ / F_{0}$ ranges from 0.1 to 10 like $Δ_{lin} / F_{0, lin}$ in (A) and (C).Reuse & Permissions
Figure 7
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ for the case of facilitation only. The parameter values are $F_{0} = 0.01$ , $Δ = 0.3$ , $r = 1 Hz$ , $τ_{F} = 80 ms$ , and $ε = 0.2$ . Circles show the results of a simulation without any approximations made (error bars within symbol size); while the solid lines illustrate the theoretical predictions expressed in Eqs. (38, 41, 34), respectively. The spectra and the coherence function show pronounced low-pass behavior. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ].Reuse & Permissions
Figure 8
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ for the case of depression only. The parameter values are $F_{0} = 0.4$ , $r = 10 Hz$ , $τ_{D} = 300 ms$ , and $ε = 0.2$ . Circles show the results of a simulation without any approximations made (error bars within symbol size); while the solid lines illustrate the theoretical predictions expressed in Eqs. (47, 50, 34). The spectra show pronounced high-pass behavior while the coherence function is perfectly flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ].Reuse & Permissions
Figure 9
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ for the general case (facilitation and depression). The parameter values are $F_{0} = 0.1$ , $Δ = 0.3$ , $r = 10 Hz$ , $τ_{F} = 300 ms$ , $τ_{D} = 100 ms$ , and $ε = 0.2$ . Circles show the results of a simulation without any approximations made (error bars within symbol size); while the solid lines illustrate the theoretical predictions. The spectra show a pronounced minimum while the coherence function is rather flat. There is only a slight deviation of the spectra between theory and simulations for low frequencies that cancel out for the coherence. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ].Reuse & Permissions
Figure 10
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ for the general case (facilitation and depression). The parameter values are $F_{0} = 0.1$ , $Δ = 0.3$ , $r = 10 Hz$ , $τ_{F} = 100 ms$ , $τ_{D} = 500 ms$ , and $ε = 0.2$ . The circles show the results of a simulation without any approximations made (error bars within symbol size); while the solid lines illustrate the theoretical predictions. The spectra show a pronounced maximum, while the simulated coherence is flat again. For the power spectrum, theory and simulation are in good accordance. However, the cross-spectrum shows a deviations for low frequencies, which leads to deviations in the coherence function. However, the coherence function as determined from simulations is rather flat. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ].Reuse & Permissions
Figure 11
(Color online) Theoretical results for (A) ${| S_{R x} (0) |}^{2} / {| S_{R x} (\infty) |}^{2}$ and (B) $S_{x x} (0) / S_{x x} (\infty)$ . The low-pass property is shown in dependence of $r τ_{F}$ and $r τ_{D}$ for a large part of the physiological parameter space. The dots with the annotations show simulation results for the ratios between the values at two different frequencies (simulation uncertainties on the left side: less than 3%; on the right side: less than 0.1%), while the values in parentheses are the corresponding theoretical results. The values of the spectra have been calculated at the frequencies 1.5 mHz and 25 Hz. They have been chosen so that according to Eqs. (53, 54, 55), the spectra do not vary much outside of this interval. The dots with the red (gray) annotations correspond to the simulations shown in Figs. 9, 10. Here, the frequency values 15 mHz and 27 Hz have been used. We simulated at constant $r = 1 Hz$ varying $τ_{F}$ and $τ_{D}$ . For theory and simulations, we used the parameter values $F_{0} = 0.1$ and $Δ = 0.3$ .Reuse & Permissions
Figure 12
(Color online) Theoretical results for the low-pass property of the coherence function $C_{R x} (f)$ expressed by the ratio of the low- and the high-frequency limit: $C_{R x} (0) / C_{R x} (\infty)$ . The low-pass property is shown in dependence of $r τ_{F}$ and $r τ_{D}$ for a large part of the physiological parameter space. The dots with the annotations show simulation results for the ratios between the values at two different frequencies (simulation uncertainties: less than 3%), while the values in parentheses are the corresponding theoretical results. The dots with the red (gray) annotations correspond to the simulations shown in Figs. 9, 10. Theory and simulation parameters are the same as in Fig. 11.Reuse & Permissions
Figure 13
(Color online) Influence of the number $N$ of synapses on the coherence function $C_{R X} (f)$ . The synapses are purely facilitating with parameters as in Fig. 7. We assume zero noise $η = 0$ . Simulations (circles) agree well with the theoretical curves (black solid lines) corresponding to Eq. (56). For $N \leq 10^{3}$ , the coherence function is proportional to $N$ preserving the low-pass property [compare Eq. (57)]. However, for $N = 10^{4}$ , we see that $C_{R X} (f)$ approaches 1 and starts to flatten out.Reuse & Permissions
Figure 14
(Color online) Influence of noise $η$ on the coherence function $C_{R X} (f)$ . We simulate $N = 10$ facilitating input synapses with $F_{0} = 0.1$ , $Δ = 0.3$ , $r = 10 Hz$ , $τ_{F} = 80 ms$ , and $ε = 0.2$ . The noise $η$ originates in 10 static synapses with amplitude $A_{Noise} = 0.4$ each of which is driven by a Poissonian process with constant rate $r_{Noise} = 10 Hz$ . The resulting coherence is well predicted by Eq. (15) (black line). In this situation, a low-pass power spectrum of the input synapses together with a flat power spectrum of the additional noise leads to a low-pass coherence function.Reuse & Permissions
Figure 15
(Color online) Influence of noise $η$ on the coherence function $C_{R X} (f)$ . We simulate $N = 10$ facilitating input synapses with parameters as in Fig. 14. The noise $η$ originates in 100 depressing synapses with $F_{0, Noise} = 0.4$ and $τ_{D, Noise} = 300 ms$ each of which is driven by a Poissonian process with constant rate $r_{Noise} = 10 Hz$ . The resulting coherence is well predicted by Eq. (15). Comparing to Fig. 14, a high-pass noise power spectrum leads to an even stronger low-pass behavior of the coherence function.Reuse & Permissions
Figure 16
(Color online) Influence of noise $η$ on the coherence function $C_{R X} (f)$ . We simulate $N = 100$ depressing input synapses with $r = 10 Hz$ , $F_{0} = 0.4$ , and $τ_{D} = 300 ms$ . The noise $η$ originates in 10 facilitating synapses with parameters as in Fig. 14; each of which is driven by a Poissonian process with constant rate $r_{Noise} = 10 Hz$ .Reuse & Permissions
Figure 17
(Color online) ${| {\hat{s}}_{R x} (f) |}^{2}$ , ${\hat{s}}_{x x} (f)$ , and $C_{R x} (f)$ . Comparison between the results of approximated and original dynamics. “simulation FD”: simulation without any approximations made; “simulation $F_{lin} D$ ”: simulation, where the approximated facilitation dynamics has been used; “simulation $F_{lin} D_{lin}$ ”: simulation, where approximated facilitation and depression dynamics has been used; “theory $F_{lin} D_{lin}$ ”: theoretical results for the approximated dynamics. The dotted coherence curve shows the coherence for the case without any synaptic plasticity [i.e., $F (t) = D (t) = 1$ ]. The parameters are the same as in Fig. 10.Reuse & Permissions

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