Influence of various temporal recoding on pavlovian eyeblink conditioning in the cerebellum

Abstract

We consider the Pavlovian eyeblink conditioning (EBC) via repeated presentation of paired conditioned stimulus (tone) and unconditioned stimulus (US; airpuff). In an effective cerebellar ring network, we change the connection probability $p_c$ from Golgi to granule (GR) cells, and make a dynamical classification of various firing patterns of the GR cells. Individual GR cells are thus found to show various well- and ill-matched firing patterns relative to the US timing signal. Then, these variously-recoded signals are fed into the Purkinje cells (PCs) through the parallel-fibers (PFs). Based on such unique dynamical classification of various firing patterns, we make intensive investigations on the influence of various temporal recoding (i.e., firing patterns) of the GR cells on the synaptic plasticity of the PF-PC synapses and the subsequent learning process for the EBC. We first note that the variously-recoded PF signals are effectively depressed by the (error-teaching) instructor climbing-fiber (CF) signals from the inferior olive neuron. In the case of well-matched PF signals, they are strongly depressed through strong long-term depression (LTD) by the instructor CF signals due to good association between the in-phase PF and the instructor CF signals. On the other hand, practically no LTD occurs for the ill-matched PF signals because most of them have no association with the instructor CF signals. This kind of “effective” depression at the PF-PC synapses coordinates firings of PCs effectively, which then makes effective inhibitory coordination on the cerebellar nucleus neuron [which elicits conditioned response (CR; eyeblink)]. When the learning trial passes a threshold, acquisition of CR begins. In this case, the timing degree $\mathcal{T}_d$ of CR becomes good due to presence of the ill-matched firing group which plays a role of protection barrier for the timing. With further increase in the number of trials, strength $\mathcal S$ of CR (corresponding to the amplitude of eyelid closure) increases due to strong LTD in the well-matched firing group, while its timing degree $\mathcal{T}_d$ decreases. In this way, the well- and the ill-matched firing groups play their own roles for the strength and the timing of CR, respectively. Thus, with increasing the number of learning trials, the (overall) learning efficiency degree $\mathcal{L}_e$ (taking into consideration both timing and strength of CR) for the CR is increased, and eventually it becomes saturated. Finally, we also discuss dependence of the variety degree for firing patterns of the GR cells and the saturated learning efficiency degree $\mathcal{L}_e$ of the CR on $p_c$ and their relations.

Appendices

Appendix

Parameter values for the LIF neuron models and the synaptic currents

In Appendix A, we list four tables which show parameter values for the LIF neuron models in Subsect. 2.3 and the synaptic currents in Subsect. 2.4. These values are adopted from physiological data (Yamazaki and Tanaka 2007; Yamazaki and Nagao 2012).

For the LIF neuron models, the parameter values for the capacitance $C_X$, the leakage current $I_L^{(X)}$, the AHP current $I_{AHP}^{(X)}$, and the external constant current $I_{ext}^{(X)}$ are shown in Table 1.

For the synaptic currents, the parameter values for the maximum conductance $\bar{g}_{R}^{(T)}$, the synaptic weight $J_{ij}^{(T,S)}$, the synaptic reversal potential $V_{R}^{(S)}$, the synaptic decay time constant $\tau _{R}^{(T)}$, and the amplitudes $A_1$ and $A_2$ for the type-2 exponential-decay function in the granular layer, the Purkinje-molecular layer, and the other parts for the CN and IO neurons are shown in Tables 2, 3, and 4, respectively.

Refined rule for synaptic plasticity

In Appendix B, we introduce a refined rule for synaptic plasticity. The coupling strength of the synapse from the pre-synaptic neuron j in the source S population to the post-synaptic neuron i in the target T population is $J_{ij}^{(T,S)}$. Initial synaptic strengths for $J_{ij}^{(T,S)}$ are given in Tables 2, 3, and 4. In this work, we assume that learning occurs only at the PF-PC synapses. Hence, only the synaptic strengths $J_{ij}^\mathrm{(PC,PF)}$ of PF-PC synapses may be modifiable (i.e., they are depressed or potentiated), while synaptic strengths of all the other synapses are static. [Here, the index j for the PFs corresponds to the two indices (M, m) for GR cells representing the mth ($1 \le m \le 50$) cell in the Mth ($1 \le M \le 2^{10}$) GR cluster.] Synaptic plasticity at PF-PC synapses have been so much studied in many experimental (Ito et al. 1982; Ito and Kano 1982; Sakurai 1987; Ito 1989; De Schutter 1995; Chen and Thompson 1995; Wang et al. 2000; Lev-Ram et al. 2003; Coesmans et al. 2004; Steuber et al. 2007; Safo and Regehr 2008; Molnár 2014; Yang and Lisberger 2014; Gallimore et al. 2018) and computational (Albus 1971; Gerstner and van Hemmen 1992; Buonomano and Mauk 1994; Kenyon et al. 1998; Medina et al. 2000a; Yamazaki and Tanaka 2007; Roberts 2007; Achard and De Schutter 2008; Yamazaki and Nagao 2012; Bouvier et al. 2018) works.

As the time t is increased, synaptic strength $J_{ij}^\mathrm{(PC,PF)}(t)$ for each PF-PC synapse is updated with the following multiplicative rule (depending on states) (Safo and Regehr 2008; Kim and Lim 2021):

$$\begin{aligned} J_{ij}^\mathrm{(PC,PF)}(t) \rightarrow J_{ij}^\mathrm{(PC,PF)}(t) + \varDelta J_{ij}^\mathrm{(PC,PF)}(t), \end{aligned}$$

(34)

where

$$\begin{aligned} \varDelta J_{ij}^\mathrm{(PC,PF)}(t)= & {} \varDelta \mathrm{LTD}_{ij}^{(1)}(t) + \varDelta \mathrm{LTD}_{ij}^{(2)}(t) + \varDelta \mathrm{LTP}_{ij}(t), \end{aligned}$$

(35)

$$\begin{aligned} \varDelta \mathrm{LTD}_{ij}^{(1)}(t)= & {} - \delta _{LTD} \cdot J_{ij}^\mathrm{(PC,PF)}(t) \cdot CF_{i}(t) \nonumber \\&\cdot \sum _{\varDelta t=0}^{\varDelta t_{r}^{*}} \varDelta J_{LTD}(\varDelta t), \end{aligned}$$

(36)

$$\begin{aligned} \varDelta \mathrm{LTD}_{ij}^{(2)}(t)= & {} - \delta _{LTD} \cdot J_{ij}^\mathrm{(PC,PF)}(t) \cdot [ 1- CF_{i}(t) ] \cdot PF_{ij}(t) \nonumber \\&\cdot D_{i}(t)\times \sum _{\varDelta t=0}^{\varDelta t_{l}^{*}} \varDelta J_{LTD} (\varDelta t), \end{aligned}$$

(37)

$$\begin{aligned} \varDelta \mathrm{LTP}_{ij}(t)= &\, {} \delta _{LTP} \cdot [J_{0}^\mathrm{(PC,PF)} - J_{ij}^\mathrm{(PC,PF)}(t)] \cdot [1-CF_{i}(t)] \nonumber \\&\cdot PF_{ij}(t)\times [1-D_{i}(t)]. \end{aligned}$$

(38)

Here, $J_{0}^\mathrm{(PC,PF)}$ is the initial value (=0.006) for the synaptic strength of PF-PC synapses. Synaptic modification (LTD or LTP) occurs, depending on the relative time difference $\varDelta t$ [= $t_\mathrm{CF}$ (CF activation time) - $t_\mathrm{PF}$ (PF activation time)] between the spiking times of the error-teaching instructor CF and the variously-recoded student PF. In Eqs. (36)-(38), $CF_i(t)$ denotes a spike train of the CF signal coming into the ith PC. When $CF_i(t)$ activates at a time t, $CF_i(t)=1$; otherwise, $CF_i(t)=0$. This instructor CF activation gives rise to LTD at PF-PC synapses in conjunction with earlier ($\varDelta t >0)$ student PF activations in the range of $t_\mathrm{CF} - \varDelta t_r^*< t_\mathrm{PF} <t_\mathrm{CF}$ ($\varDelta t_r^* \simeq 277.5$ msec), which corresponds to the major LTD in Eq. (36).

We next consider the case of $CF_i(t)=0$, which corresponds to Eqs. (37) and (38). Here, $PF_{ij}(t)$ denotes a spike train of the PF signal from the jth pre-synaptic GR cell to the ith post-synaptic PC. When $PF_{ij}(t)$ activates at time t, $PF_{ij}(t)=1$; otherwise, $PF_{ij}(t)=0$. In the case of $PF_{ij}(t)=1$, PF firing may cause LTD or LTP, depending on the presence of earlier CF firings in an effective range. If CF firings exist in the range of $t_\mathrm{PF} + \varDelta t_l^*< t_\mathrm{CF} <t_\mathrm{PF}$ ($\varDelta t_l^* \simeq -117.5$ msec), $D_i(t)=1$; otherwise $D_i(t)=0$. When both $PF_{ij}(t)=1$ and $D_i(t)=1$, the PF activation causes another LTD at PF-PC synapses in conjunction with earlier ($\varDelta t <0$) CF activations [see Eq. (37)]. The probability for occurrence of earlier CF firings within the effective range is very low because mean firing rates of the CF signals (corresponding to output firings of individual IO neurons) are $\sim$ 1.5 Hz (Mathy et al. 2009; Llinás 2014). Hence, this 2nd type of LTD is a minor one. In contrast, in the case of $D_i(t)=0$ (i.e., absence of earlier associated CF firings), LTP occurs because of the PF firing alone [see Eq. (38)]. The update rate $\delta _{LTD}$ for LTD in Eqs. (36) and (37) is 0.005, while the update rate $\delta _{LTP}$ for LTP in Eqs. (38) is 0.0005 (=$\delta _{LTD}/10$) (Yamazaki and Nagao 2012).

Table 5 List of abbreviations

In the case of LTD in Eqs. (36) and (37), the synaptic modification $\varDelta J_{LTD} (\varDelta t)$ changes depending on the relative time difference $\varDelta t$ $(= t_\mathrm{CF} - t_\mathrm{PF}$). We use the following time window for the synaptic modification $\varDelta J_{LTD} (\varDelta t)$ (Safo and Regehr 2008; Kim and Lim 2021):

$$\begin{aligned} \varDelta J_{LTD}(\varDelta t) = A + B \cdot e^{-(\varDelta t -t_0)^2/\sigma ^2}, \end{aligned}$$

(39)

where $A=-0.12$, $B=0.4$, $t_0 = 80$, and $\sigma =180$. The time window for $\varDelta J_{LTD} (\varDelta t)$ is well shown in Fig. 3 in Ref. (Kim and Lim 2021), where LTD occurs in an effective range of $\varDelta t_l^*< \varDelta t < \varDelta t_r^*$. We note that a peak appears at $t_0=80$ msec, and hence peak LTD takes place when PF firing precedes CF firing by 80 msec. A CF firing gives rise to LTD in association with earlier PF firings in the black region ($0< \varDelta t < \varDelta t_r^*$), and it also causes to another LTD in conjunction with later PF firings in the gray region ($\varDelta t_l^*< \varDelta t <0$). The effect of CF firing on earlier PF firings is much larger than that on later PF firings. However, outside the effective range (i.e., $\varDelta t > \varDelta t_r^*$ or $< \varDelta t_l^*$), PF firings alone results in occurrence of LTP, because of absence of effectively associated CF firings.

Our refined rule for synaptic plasticity has the following advantages for the $\varDelta \mathrm{LTD}$ in comparison with that in (Yamazaki and Tanaka 2007; Yamazaki and Nagao 2012). Our rule is based on the experimental result in (Safo and Regehr 2008). In the presence of a CF firing, a major LTD ($\varDelta \mathrm{LTD}^{(1)}$) occurs in conjunction with earlier PF firings in the range of $t_\mathrm{CF} - \varDelta t_r^*< t_\mathrm{PF} <t_\mathrm{CF}$ ($\varDelta t_r^* \simeq 277.5$ msec), while a minor LTD ($\varDelta \mathrm{LTD}^{(2)}$) takes place in conjunction with later PF firings in the range of $t_\mathrm{CF}< t_\mathrm{PF} <t_\mathrm{CF} - \varDelta t_l^*$ ($\varDelta t_l^* \simeq -117.5$ msec). The magnitude of LTD varies depending on $\varDelta t$ (= $t_\mathrm{CF}$ - $t_\mathrm{PF}$); a peak LTD occurs when $\varDelta t =80$ msec. In contrast, the rule in (Yamazaki and Nagao 2012; Yamazaki and Tanaka 2007)considers only the major LTD in association with earlier PF firings in the range of $t_\mathrm{CF} - 50< t_\mathrm{PF} <t_\mathrm{CF}$, the magnitude of major LTD is equal, independently of $\varDelta t$, and minor LTD in conjunction with later PF firings is not considered. Outside the effective range of LTD, PF firings alone lead to LTP in both rules. However, we also note that some features of Pavlovian EBC were successfully reproduced by using the simple synaptic rule with only the major LTD in (Yamazaki and Tanaka 2007).

List of abbreviations

In Appendix C, we present a list of abbreviations which is shown in Table 5.