Abstract

It is well known that cerebellar motor control is fine-tuned by the learning process adjusted according to rich error signals from inferior olive (IO) neurons. Schweighofer and colleagues proposed that these signals can be produced by chaotic irregular firing in the IO neuron assembly; such chaotic resonance (CR) was replicated in their computer demonstration of a Hodgkin-Huxley (HH)-type compartment model. In this study, we examined the response of CR to a periodic signal in the IO neuron assembly comprising the Llinás approach IO neuron model. This system involves empirically observed dynamics of the IO membrane potential and is simpler than the HH-type compartment model. We then clarified its dependence on electrical coupling strength, input signal strength, and frequency. Furthermore, we compared the physiological validity for IO neurons such as low firing rate and sustaining subthreshold oscillation between CR and conventional stochastic resonance (SR) and examined the consistency with asynchronous firings indicated by the previous model-based studies in the cerebellar learning process. In addition, the signal response of CR and SR was investigated in a large neuron assembly. As the result, we confirmed that CR was consistent with the above IO neuron’s characteristics, but it was not as easy for SR.

1  Introduction

Over the past few decades, numerous studies have been conducted on cerebellar function (Marr, 1969; Albus, 1971; Ito, Sakurai, & Tongroach, 1982; Llinás, Leznik, & Makarenko, 2002; Kitazawa & Wolpert, 2005; Lang et al., 2016). It is now well known that the cerebellum is the motor control system that allows for accurate and coordinated movements. Purkinje cells in the cerebellum receive two kinds of synaptic input. The first is more than 100,000 parallel fibers, and the second is a single climbing fiber as an axon from an inferior olive (IO) neuron; these inputs evoke “simple spikes” as motor control signals and “complex spikes” including Purkinje cell error signals, respectively. In the cerebellar learning theory, motor control function is obtained during a learning process where synaptic weights between the parallel fibers and Purkinje cells are adjusted by long-term depression (LTD) according to the error signal from IO neurons (Marr, 1969; Albus, 1971; Ito et al., 1982).

The IO neuron has some electrophysiological features that have been recorded experimentally, such as sinusoidal subthreshold oscillations in membrane potential and spike generation at two thresholds (Latham & Paul, 1971; Llinas & Yarom, 1981a, 1981b, 1986). Schweighofer et al. insisted that these behaviors are caused by the difference of membrane potentials between the soma and dendrites. To describe IO neuron dynamics, they proposed a Hodgkin-Huxley (HH) type of compartment system composed of the soma and dendrites, focusing on their detailed ion channel structure obtained experimentally (Schweighofer, Doya, & Kawato, 1999). Llinás et al., (2002) constructed a simpler model that consists of systems with empirically observed modes of the above membrane potential dynamics (referred to as the Llinás approach IO neuron model from now on) (Velarde, Nekorkin, Kazantsev, Makarenko, & Llinás, 2002). Using similar types of this model, several studies have shown that IO networks maintain motor execution patterns with robustness and exhibit the prompt stimulus-induced phase reset by the chaotic behavior (Kazantsev, Nekorkin, Makarenko, & Llinás, 2003, 2004; Maslennikov & Nekorkin, 2012). These are from the viewpoint of the motor clock theory (also called cerebellar timing theory) where IO neurons do not provide the error signal for learning but the periodic clock for coordinating movements (Llinás & Welsh, 1993; Keating & Thach, 1995; Welsh & Llinás, 1996; Llinás et al., 2002).

In cerebellar learning theory, regarding IO neuron signal transmission efficiency, each error signal must carry a sufficient amount of information for cerebellar learning. However, the IO neuron firing rate is actually very low (below 1 Hz), and IO firings are synchronous due to extensive electrical coupling (Llinas & Yarom, 1981a, 1981b, 1986; Llinás & Welsh, 1993; Keating & Thach, 1995; Welsh & Llinás, 1996; Blenkinsop & Lang, 2006). It has been recognized that this feature apparently contradicts the error signal with high temporal resolution (Kobayashi et al., 1998; Gilbert & Thach, 1977). A simulation study by Schweighofer et al. (2004) showed that chaotic irregular and asynchronous firings could arise in the network consisting of their IO neuron model with realistic moderate electrical coupling strength. The firings generated at different timings against each trial enabled the network to transmit rich error signal through the trial average. This result is supported by the experimental evidence, although indirectly (Kitazawa & Wolpert, 2005; Kobayashi et al., 1998). In recent years, it has been found that the cerebellar learning system comprising IO neuron, Purkinje cells, and cerebellar nuclei can adjust electrical coupling strength (Best & Regehr, 2009). Several simulation studies have shown that this mechanism enables enhancement of the signal response by shifting the spiking patterns (i.e., from synchronous and periodic spiking to asynchronous and chaotic spiking (Kawato, Kuroda, & Schweighofer, 2011; Schweighofer, Lang, & Kawato, 2013; Tokuda, Hoang, Schweighofer, & Kawato, 2013). This phenomenon is not conventional stochastic resonance (SR) (Wiesenfeld & Moss, 1995; Moss & Wiesenfeld, 1995; Gammaitoni, Hänggi, Jung, & Marchesoni, 1998) by noise; rather, it is the signal response enhanced by chaos, which is called chaotic resonance (CR) (Carroll & Pecora, 1993; Crisanti, Falcioni, Paladin, & Vulpiani, 1994; Nicolis, Nicolis, & McKernan, 1993; Sinha & Chakrabarti, 1998; Anishchenko, Neiman, & Safanova, 1993; Nishimura, Katada, & Aihara, 2000; Anishchenko et al., 2007; Nobukawa, Nishimura, & Katada, 2012; Nobukawa, Nishimura, Yamanishi, & Liu, 2015).

In the CR phenomenon under the condition where no additive noise exists, the system responds to the weak input signal by intrinsic chaotic activities. At first, its characteristic was investigated with simple models such as the one-dimensional cubic map and Chua’s circuit (Carroll & Pecora, 1993; Crisanti et al., 1994; Nicolis et al., 1993; Sinha & Chakrabarti, 1998; Anishchenko, Neiman, & Safanova, 1993). Recently, CR has been studied in neural systems including IO neural systems (Schweighofer et al., 2004; Tokuda et al., 2013). However, few studies have described the fundamental properties of CR (i.e., the characteristics against input signal amplitude and frequency) or compared it with conventional SR in IO neuron models. In addition neuron assemblies in previous studies included fewer than 100 neurons (Schweighofer et al., 2004; Tokuda et al., 2013). Therefore, CR has not been investigated in large neuron assemblies approaching the real size of mammalian IO neuron assembly—roughly tens of thousands of neurons in cats (Brodal & Kawamura, 2012). Under the mutual interactions among neurons with the electrical coupling, the expansion of network size may have the potential to affect the characteristic of signal response.

We previously examined the characteristics of CR in a single IO neuron using the Llinás approach model from the viewpoint of motor clock theory but cerebellar learning theory and compared the physiological validities of IO for CR and SR (Nobukawa & Nishimura, 2013b). We have also attempted to investigate CR in the neuron assembly (Nobukawa, Nishimura & Katada, 2011; Nobukawa & Nishimura, 2013a). Based on the outcomes of our research conducted in the single IO neuron (Nobukawa & Nishimura, 2013b), here we examine the response of CR to a periodic signal in the coupled Llinás approach IO neuron assembly and clarify its dependence on electrical coupling strength, input signal strength, and frequency. We then compare the physiological validity of IO neurons such as low firing rate and sustaining subthreshold oscillation between CR and SR and examine the consistency with asynchronous firings indicated by the previous model-based studies for the cerebellar learning process (Kawato et al., 2011; Schweighofer et al., 2013; Tokuda et al., 2013). Furthermore, the signal response of CR is investigated in a large neuron assembly.

2  Model and Methods

2.1  The Llinás Approach IO Neuron Model and Its Interaction

The Llinás approach IO neuron model is a system composed of a Van der Pol (VP) subsystem, high-threshold FitzHugh-Nagumo (FNI) subsystem, and low-threshold FitzHugh-Nagumo (FNII) subsystem to reproduce experimentally observed membrane potential behaviors such as subthreshold oscillations and spike generation at two different thresholds (Llinas & Yarom, (1986). Figure 1 shows that each subsystem is coupled by parameters , and , when the coupling strength between VP and FNI is set to 1 (normalized). This model is described as follows:
formula
2.1
formula
2.2
formula
2.3
where and are variables of VP, FNI, and FNII, respectively. Note that corresponds to the membrane potential. Here, the functions of and that determine the characteristic of threshold in FN subsystems are given by
formula
2.4
formula
2.5
where and determine the high and low thresholds, respectively. As shown in Figure 2, this model can reproduce the major spike patterns observed experimentally, such as subthreshold oscillation (see Figure 2a), spiking at a high threshold when the membrane is depolarized (see Figure 2b), spiking at a low threshold when the membrane is hyperpolarized (see Figure 2c), and spiking at high and low thresholds (see Figure 2d) (Nobukawa & Nishimura, 2013b).
Figure 1:

Constitution of the Llinás approach IO neuron model. VP: Van der Pol subsystem. FNI: Low-threshold FitzHugh-Nagumo subsystem. FNII: High-threshold FitzHugh-Nagumo subsystem.

Figure 1:

Constitution of the Llinás approach IO neuron model. VP: Van der Pol subsystem. FNI: Low-threshold FitzHugh-Nagumo subsystem. FNII: High-threshold FitzHugh-Nagumo subsystem.

Figure 2:

Typical behaviors of membrane potential and the internal variable . (a) . (b) (c) . (d) ).

Figure 2:

Typical behaviors of membrane potential and the internal variable . (a) . (b) (c) . (d) ).

In our simulations, we adopt two types of assembly similar to those used in (Schweighofer et al., 2004) as shown in Figure 3. The type in Figure 3a is the chain assembly of 10 neurons in which each neuron couples its two nearest neighbors. The type in Figure 3a is the lattice assembly of neurons in which each neuron couples its four nearest neighbors. Here, each neuron couples through FNII subsystems having the membrane potential with a gap junction described as the difference among the membrane potentials of coupled neurons. In the chain assembly, the equation for in equation (2.3) is replaced by
formula
2.6
where represents the membrane potential of the th neuron () and is the coupling strength of the gap junctions. In the lattice assembly, the equation for becomes
formula
2.7
where represents the membrane potential of the th neuron () and , and are the membrane potentials of four nearest neighbors. Here the periodic boundary condition is applied to these assemblies as shown in Figure 3.
Figure 3:

IO neuron assembly configuration.

Figure 3:

IO neuron assembly configuration.

To evaluate the signal response of the Llinás approach IO neuron assembly, we consider a weak periodic signal and a gaussian white noise () for 2.6 and 2.7 as follows:
formula
2.8
Here, is a typical example of the error signal applied to all neurons for motor learning. In the case of CR, noise is absent (), and in the case of SR, noise is applied ().

2.2  Evaluation Indices

In this section, we introduce mutual correlation , mutual information , and conforming ability to evaluate the response of the IO neuron assembly to the signal . In addition, other indices (e.g., coherence measure , firing frequency to the subthreshold oscillation frequency ratio , correlation time , maximum Lyapunov exponent ) are used to check the neuron assembly state.

We evaluate the timing of spikes against the signal by using the cycle histogram . is a histogram of firing counts at mod () against the signal with a period , . For example, in the case of the spike times , the values of mod are 21, 66, 21, 66, 66, (), and then the cycle histogram becomes and . In our simulation, we set the width of the bin of the cycle histogram to and used ms as the evaluation duration.

To quantify the signal response, we used the following indices. The mutual correlation between the cycle histogram of the neuron spikes and the signal is given by
formula
2.9
formula
2.10
where indicates average in . For the time delay factor , we check (i.e., the largest between ).
As an extensively used index for evaluating the information transmission, we employ mutual information that is transmitted from input to output :
formula
2.11
Here, and consist of and event states {}, {} into which () and ( its maximum value) are divided equally, respectively. and are given by
formula
2.12
formula
2.13
where and are the occurrence probabilities. These are defined by the occurrence rate for the event states assuming different values against , that is, counting the event states from to in 50 bins of . is the conditional probability for and . In our simulation, we set , but if the maximum value of is smaller than , is set to the maximum value because is an integer of firing counts.
The coherence measure (Wang & Buzsáki, 1996) is utilized as an index of spike synchronization in the assembly. Coherence between two neurons and is measured by the following equation:
formula
2.14
is the spike train of the th neuron during a long-term interval and given by 0 or 1. Here, if the th neuron fires within the th time bin, ; otherwise, . for the th neuron is given in the same way. In the simulation, we take and . The population coherence of the entire assembly is obtained by the average of overall neuron pairs,
formula
2.15
where is the number of neurons in the assembly.
Conforming ability to the signal is measured by the strength of the interspike interval distribution around the input period (Sinha & Chakrabarti, 1998):
formula
2.16
Here, indicates the probability distribution of the interspike interval . In the simulation, we use during to evaluate . In contrast with and by the cycle histogram, removes the influence of the spikes lagged over . By this effect, the frequency characteristics can stand out. Therefore, it has been used for evaluating frequency characteristics in SR and CR (Sinha & Chakrabarti, 1998; Nobukawa et al., 2012; Nobukawa & Nishimura, 2013b).
To evaluate the physiological validity of the firing rate, we introduce firing frequency to the subthreshold oscillation frequency ratio:
formula
2.17
where and indicate the firing frequency and the subthreshold oscillation frequency, respectively. Because the firing frequency of an IO neuron is normally around or lower than 1 Hz and the subthreshold oscillation is about 10 Hz, becomes smaller than 0.1 (Latham & Paul, 1971). Even if an IO neuron spikes at every subthreshold oscillation, does not exceed the limit of 1.0. Therefore, we determine as the physiological range of .
To measure the periodicity of the membrane potential , we adopt the correlation time
formula
2.18
for each neuron (Pikovsky & Kurths, 1997). Here, and indicate the autocorrelation of on time delay and 20 periods of subthreshold oscillation, respectively. Then the values of are averaged among all neurons as . In this study, we adopt 0.3 as the allowable lower limit of (see the appendix).
To evaluate chaos in the system, we use the maximum Lyapunov exponent (Parker & Chua, 2012):
formula
2.19
Here, () are perturbed initial conditions applied at , and are their time evolution for .

2.3  Setting for Model Parameters

For evaluating the signal response in CR, we search the parameter region where the system state becomes chaotic. Here, we use the parameter set of subthreshold oscillation in Figure 2a as the starting point for acquiring the chaotic parameter region. The parameters and are fixed. Then we investigate the dependence of on the remaining parameters and , which couple the FNII with VP and , which couples the FNI with FNII, as shown in Figure 1. Figure 4 shows the dependence of on and in the case of . The chaotic region () is widely distributed in . Next, the dependence of on is investigated as shown in Figure 5 under the condition and . It is confirmed that the chaotic region exists in . From these results, the and are kept to 0.95 and 0.9, respectively, throughout our simulation in both CR and SR cases.

Figure 4:

Dependence of maximum Lyapunov exponent on and . .

Figure 4:

Dependence of maximum Lyapunov exponent on and . .

Figure 5:

Order parameter dependence of maximum Lyapunov exponent . .

Figure 5:

Order parameter dependence of maximum Lyapunov exponent . .

3  Results and Discussion

3.1  Chaotic Behavior in the Llinás Approach IO Neuron Model

3.1.1  Chaotic Behavior in a Single Neuron

We studied the chaotic characteristic of a single IO neuron by using the Llinás approach model as follows (Nobukawa & Nishimura, 2013b). Figure 6 indicates the magnified peak of Figure 5 in . In this region, the system has a chaotic state () in and a periodic state () on both sides of the chaotic region. The behaviors of time series corresponding to the values shown by black points in Figure 6 are demonstrated in Figure 7. In each value, the system state becomes periodic spiking (Figure 7a) , chaotic spiking (Figure 7b) ), subthreshold chaotic oscillation (Figure 7c), , and subthreshold periodic oscillation (Figure 7d, ), respectively.

Figure 6:

Order parameter dependence of maximum Lyapunov exponent around the chaotic region. .

Figure 6:

Order parameter dependence of maximum Lyapunov exponent around the chaotic region. .

Figure 7:

Behaviors of time series at . (a) Periodic spikes, . (b) Chaotic spikes, . (c) Subthreshold chaotic oscillation, . (d) Subthreshold periodic oscillation, shown by black points in Figure 6.

Figure 7:

Behaviors of time series at . (a) Periodic spikes, . (b) Chaotic spikes, . (c) Subthreshold chaotic oscillation, . (d) Subthreshold periodic oscillation, shown by black points in Figure 6.

3.1.2  Chaotic Behavior in Neuron Assemblies

Let us demonstrate the behaviors of neuron assemblies consisting of the periodic spiking neuron given by Figure 7a . Figure 8 indicates the raster plot for neuron spikes of the chain and lattice assembly types. As shown in Figure 8a, since is too weak for each neuron to interact, each neuron spikes periodically and asynchronously in the chain and lattice assembly types (). As the coupling strength is increased such as in the chain type and in the lattice type, the spiking in the neuron assemblies becomes chaotic and asynchronous (; see Figure 8b). The further strong coupling strength such as (chain type) and 1 (lattice type) induces periodic and synchronous spiking (; see Figure 8c). Recently it was revealed that signal feedback from a cerebellar nucleus to an IO nucleus can change the electrical coupling strength among IO neurons (Best & Regehr, 2009). The simulation studies have shown that this mechanism permits IO to change the spiking pattern between periodic and synchronous mode and nonperiodic and asynchronous mode (Kawato et al., 2011; Schweighofer et al., 2013; Tokuda et al., 2013). These findings correspond to the dependence of the spiking pattern on the electrical coupling strength given by Figure 8.

Figure 8:

Raster plot in chain- and lattice-type assemblies. (a) Asynchronous and periodic spiking in the case of . (b) Asynchronous and chaotic spiking in the case of (chain type) and (lattice type). (c) Synchronous and periodic spiking in the case of (chain type) and (lattice type).

Figure 8:

Raster plot in chain- and lattice-type assemblies. (a) Asynchronous and periodic spiking in the case of . (b) Asynchronous and chaotic spiking in the case of (chain type) and (lattice type). (c) Synchronous and periodic spiking in the case of (chain type) and (lattice type).

3.2  Fundamental Properties of Chaotic Resonance

3.2.1  Dependence on Electrical Coupling Strength

This section concerns the deterministic response of the system in the noise-free case (). The parameter is taken as , where the membrane potential spikes periodically when the IO neurons are uncoupled (). Figure 9 shows the time series of , the first neuron of the coupled assembly and the corresponding cycle histogram of firing counts under a weak signal strength (). In Figure 9a, , the neurons fire periodically unrelated to the signal , and the cycle histogram does not respond to the signal in either the chain or lattice type. However, by increasing coupling strength to an appropriate value, the neurons begin to fire nonperiodically, and the cycle histogram fits the signal with some time delay as shown in Figure 9b, . Beyond the appropriate region of , the assembly loses the signal response.

Figure 9:

Time series of membrane potential (left) and corresponding cycle histogram (right) under the sinusoidal input signal (dotted line). (a) For , (b) for in chain and lattice types. .

Figure 9:

Time series of membrane potential (left) and corresponding cycle histogram (right) under the sinusoidal input signal (dotted line). (a) For , (b) for in chain and lattice types. .

Let us examine the dependence of the evaluation indices for the above data. Figure 10 indicates dependence of , , , , and . The and peak near 0.9 and 1.3 at around , respectively (see Figures 10a and 10b). In this region, the firing rate is low () (see Figure 10d), the neurons in the assembly fire asynchronously () (see Figure 10c), and the assembly has a chaotic state () (see Figure 10d).

Figure 10:

Dependence of signal response on coupling strength in the CR case. (a) Mutual correlation between and . (b) Mutual information between and . (c) Coherence measure . (d) Firing frequency to subthreshold osculation frequency ratio (). The red shaded area indicates the physiological range of (). (e) Maximum Lyapunov exponent . .

Figure 10:

Dependence of signal response on coupling strength in the CR case. (a) Mutual correlation between and . (b) Mutual information between and . (c) Coherence measure . (d) Firing frequency to subthreshold osculation frequency ratio (). The red shaded area indicates the physiological range of (). (e) Maximum Lyapunov exponent . .

From these results, it is found that the appropriate coupling strength leads the assembly to the asynchronous chaotic firing state, and the response to the signal is enhanced due to this activity. This signal response can be interpreted as CR.

3.2.2  Dependence on Signal Strength

We also examine the dependence on the signal strength in CR. Figure 11 shows dependence of , , , , and under the condition of fixing values that are (chain type) and (lattice type), respectively. At , signal response efficiency is low ( and ), but it increases accompanied by signal strength , and and have the peak at , maintaining a low firing rate (), asynchronous firing (), and a chaotic state (). However, further enhancing signal strength , periodic and synchronous spikes (, ) decrease and . From these results, it is found that CR arises under the condition of weak signal strength where the system maintains the chaotic state. In the recent model-based studies about the cerebellar learning process, a chaotic spiking pattern is observed only under the condition that the error signal becomes weak in the later learning stage (Kawato et al., 2011; Schweighofer et al., 2013; Tokuda et al., 2013). Thus, the simulation result in this section is consistent with these findings.

Figure 11:

Dependence of signal response on signal strength in the case of CR. (a) . (b) . (c) . (d) . (e) . ( (lattice) (chain)).

Figure 11:

Dependence of signal response on signal strength in the case of CR. (a) . (b) . (c) . (d) . (e) . ( (lattice) (chain)).

3.2.3  Dependence on Signal Frequency

Next, we evaluate the dependence on the signal frequency in CR under the condition ( (chain type) and (lattice type), ). As shown in Figure 12, the signal frequency dependence of has a peak at with the chaotic state (). Thus, CR has the resonance frequency as observed for general resonance phenomena.

Figure 12:

Dependence of signal response on signal frequency in the case of CR. (a) in 2.16. (b) . ( (lattice) (chain)).

Figure 12:

Dependence of signal response on signal frequency in the case of CR. (a) in 2.16. (b) . ( (lattice) (chain)).

3.3  Comparison with Conventional SR

In this section, we examine the conventional SR where the noise () and input signal () are applied and compare the characteristics of spike synchronization, firing frequency, and subthreshold oscillation with those of CR. Figure 13 shows the scatter plot of and in SR and CR cases in the chain-type (see Figure 13a) and lattice- (see Figure 13b) type assemblies. Regarding SR, the open circles indicate the 28 points of chain case in Figure 13a and 28 points of lattice case in Figure 13b surviving of the 1440 combinations of , , and (here, is divided into 10 in each order) under the following conditions: high signal response efficiency (), sustained low firing frequency (), and being the periodic () and no-firing state if the noise is not applied. In Figures 13a and 13b, the points that fulfill are three points of , , and two points of , respectively.

Figure 13:

Scatter plot of and in SR and CR cases under the condition of high signal response efficiency () and sustained low firing frequency (). The red shaded area indicates the physiological range of (). (a) Chain type with 10 neurons. (b) Lattice type with neurons. CR: . SR: .

Figure 13:

Scatter plot of and in SR and CR cases under the condition of high signal response efficiency () and sustained low firing frequency (). The red shaded area indicates the physiological range of (). (a) Chain type with 10 neurons. (b) Lattice type with neurons. CR: . SR: .

Next, we examined the signal response dependence on the noise strength in detail for the above five points as shown in Figure 14. Figures 14a to 14d indicate that both and for all five points have peaks at . Around , their (Figures 14e and 14f) and (Figures 14g and 14h) take and , (i.e., the systems are barely sustaining the subthreshold oscillations and asynchronous firing). However, the range of to fulfill low firing frequency () is confined in that smaller than . In addition, we examine the signal response dependence of the signal frequency for the above five points as shown in Figure 15. In cases for chain-type and lattice-type assemblies in, respectively, (Figures 15a and 15b), the signal frequency dependences of exhibit the peaks at whose values ( in both types assemblies) are lower in comparison with CR in Figure 12. That is, SR in this condition barely has resonance frequency. However, in cases for both types of assemblies, the dependencies of lose the peaks due to decreasing more from value for the peaks of and ().

Figure 14:

Noise strength dependence of the signal response of SR in chain-type (left) and lattice-type (right) assemblies. (a, b) For . (c, d) For . (e, f) For . (g, h) For . (i, j) For . .

Figure 14:

Noise strength dependence of the signal response of SR in chain-type (left) and lattice-type (right) assemblies. (a, b) For . (c, d) For . (e, f) For . (g, h) For . (i, j) For . .

Figure 15:

Dependence of signal response on signal frequency in the case of SR. (a) in chain-type assemblies. (b) in lattice-type assemblies. .

Figure 15:

Dependence of signal response on signal frequency in the case of SR. (a) in chain-type assemblies. (b) in lattice-type assemblies. .

Regarding CR, all five points in the chain type and four points in the lattice type that fulfill and in Figure 10, distribute in and , as shown in Figures 13a and 13b, respectively. Therefore, the region where CR arises meets the conditions for the physiological validity of IO neuron assemblies such as low firing frequency and sustaining subthreshold oscillation and the consistency with asynchronous firing indicated by the previous model-based studies in the cerebellar learning process (Kawato et al., 2011; Schweighofer et al., 2013; Tokuda et al., 2013). This is quite a contrast to the difficulty of the region where SR arises. This result is similar to the tendency of the single IO neuron case in our previous study (Nobukawa & Nishimura, 2013b).

3.4  CR and SR in a Large Neuron Assembly

In this section, we examine the signal response of CR in a larger set of assemblies whose size expands from 10 and neurons to a realistic IO neuron assembly size (Brodal & Kawamura, 2012). Figure 16a shows the raster plot for neuron spikes in the case of applying with to chain-type and lattice-type neuron assemblies where and neurons are coupled by , respectively. The other parameters are set to . In this case, the state of neuron assemblies becomes periodic () and asynchronous (), and the neurons spike autonomously, not depending on the period of input signal . However, in the condition of intermediate coupling strength for chain and for lattice, the state becomes chaotic () and more asynchronous (), and the period between the bands of high spike density corresponds with , as shown in Figure 16b. Thus, CR also seems to arise in larger neuron assemblies and meets the conditions for real IO neurons.

Figure 16:

Raster plot in chain- and lattice-type neuron assemblies consisting of and neurons, respectively. (a) Each neuron is coupled by weak coupling strength . (b) By intermediate coupling strength for chain and for lattice. .

Figure 16:

Raster plot in chain- and lattice-type neuron assemblies consisting of and neurons, respectively. (a) Each neuron is coupled by weak coupling strength . (b) By intermediate coupling strength for chain and for lattice. .

Concerning the above signal response, the dependence of the evaluation indices on is examined and compared with the result in the cases of small neuron assemblies, as already noted. As shown in Figure 17, by the effect of CR, and have 1.0 (see Figure 17a) and 1.3 (see Figure 17b) at around , respectively, in the chaotic firing region ( in Figure 17e). In this region, the physiological validity for IO neurons, i.e., low firing rate ( in Figure 17d) and sustained sub-threshold oscillation ( in Figure 17f) and the consistency with the previous model-based studies in cerebellar learning process (Kawato et al., 2011; Schweighofer et al., 2013; Tokuda et al., 2013), i.e., asynchronous firing ( in Figure 17c), are satisfied. In comparison with the small neuron assemblies in Figure 10, the high signal response region with expands from (10-neuron case) to ( neuron case) in the chain type and from (–neuron case) to (–neuron case) in the lattice type. This is supported by the corresponding expansion of the chaotic firing region as shown in Figure 17 e. Similar tendencies are also observed for .

Figure 17:

Dependence of CR signal response on coupling strength in neuron assemblies consisting of (chain type) and (lattice type) neurons. (a) . (b) . (c) . (d) . (e) . (f) . .

Figure 17:

Dependence of CR signal response on coupling strength in neuron assemblies consisting of (chain type) and (lattice type) neurons. (a) . (b) . (c) . (d) . (e) . (f) . .

In addition, Figure 18 shows the effect of neuron assembly size on the scatter plots of and for CR (see Figures 18a and 18b) and SR (see Figures 18c and 18d). Here, the data points in these scatter plots indicate the results in 10, , , , , and neuron assembly cases. They fulfill the same conditions used in Figure 13, that is, and , and being the periodic () and no-firing state if the noise is not applied in SR cases. All points in CR cases satisfy and , while the points in SR cases satisfy this condition only at as with the small size of neuron assembly cases. This means that CR sustains the above-mentioned characteristics for the IO neurons in both small and larger neuron assemblies, but SR barely sustains.

Figure 18:

Scatter plot of and for CR and SR under the condition of high signal response efficiency () sustaining a low firing frequency () in the chain-type assemblies consisting of 10, , neurons and the lattice-type assemblies consisting of , , neurons. (a) CR cases in the chain type. (b) CR cases in the lattice type. (c) SR cases in the chain type. (d) SR cases in the lattice type. CR: SR: .

Figure 18:

Scatter plot of and for CR and SR under the condition of high signal response efficiency () sustaining a low firing frequency () in the chain-type assemblies consisting of 10, , neurons and the lattice-type assemblies consisting of , , neurons. (a) CR cases in the chain type. (b) CR cases in the lattice type. (c) SR cases in the chain type. (d) SR cases in the lattice type. CR: SR: .

Finally, we evaluate the dependence of the signal response on neuron assembly size in the situation of a much shorter duration than used for constructing the in Figure 9. Figures 19a and 19b show the dependencies of and on neuron assembly size, respectively. In both cases, their increases correspond with the size of neuron assembly and show the tendency of saturating to and at around neurons. The neuron assemblies sustain the chaotic states () in all size cases, as shown in Figure 19c. These findings show that the evaluation time to attain high signal response efficiency in CR can be shortened due to the increased size of the neuron assembly.

Figure 19:

Dependence of signal response on neuron assembly size for CR. (a) . (b) . (c) . (chain type) (lattice type).

Figure 19:

Dependence of signal response on neuron assembly size for CR. (a) . (b) . (c) . (chain type) (lattice type).

4  Conclusion

In this study, we examined the fundamental properties of CR in IO neuron assemblies concerning its dependence on electrical coupling strength, input signal strength, and frequency. We compared the signal responses of CR and SR with respect to the physiological validity for IO neurons such as IO neurons' low firing rate and sustaining subthreshold oscillation and examined the consistency with asynchronous firing among IO neurons indicated by the previous model-based studies in the cerebellar learning process. Furthermore, the influence on the CR signal response has been investigated in circumstances where the neuron assembly’s size is expanded to around neurons.

As a result, CR was found to arise with appropriate coupling strength for the weak signal and have the frequency response characteristic as with general resonance phenomena. Moreover, we confirmed that CR was consistent with the above characteristics of IO neuron assembly, but it was not as easy for SR to meet these characteristics in the allowed regions where SR arises. This finding suggests that CR functions more effectively on the IO neuron assembly during the cerebellar learning compared to SR. Finally, the high signal response region for the coupling strength in CR was revealed to expand in the larger neuron assemblies, although the above characteristics were identical to those in the small neuron assemblies. Our results also show that the evaluation time to attain high signal response efficiency is shortened due to the increased size of the neuron assembly. In future work, we plan to examine signal responses in a cerebellar-supervised learning system like a triangle circuit comprising an IO neuron assembly, Purkinje cells, and cerebellar nuclei.

Appendix

We investigated the allowable range of in SR by observing the time series of and the corresponding . Figure 20 shows the dependence of in the case of and . In , the value of keeps a comparable level () for the noise-free condition. However, the value of decreases from and becomes 0 at . Then the time series of in the case of , given by solid circles in Figure 20, is demonstrated in Figure 21. At , the shape of the subthreshold oscillation in Figure 21a is maintained, but it is disturbed with increasing , as shown in Figures 21b to 21d). The limit of disturbed shape allowed physiologically is that around , and its corresponding is approximately 0.38. So, we adopt 0.3 as the lower limit of .

Figure 20:

Noise strength dependence of in the chain-type assembly. .

Figure 20:

Noise strength dependence of in the chain-type assembly. .

Figure 21:

Behaviors of time series at (a) , (b) , (c) , and (d) shown by solid circles in Figure 20. .

Figure 21:

Behaviors of time series at (a) , (b) , (c) , and (d) shown by solid circles in Figure 20. .

Acknowledgments

This work is supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B), grant number (15K21471).

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