Bodily motion fluctuation improves reaching success rate in a neurophysical agent via geometric-stochastic resonance
Bodily motion fluctuation improves reaching success rate in a neurophysical agent via geometric-stochastic resonance
Shogo Yonekura 0 1
Yasuo Kuniyoshi 0 1
0 The University of Tokyo , Tokyo , Japan
1 Editor: Yong Deng, Southwest University , CHINA
Organisms generate a variety of noise types, including neural noise, sensory noise, and noise resulting from fluctuations associated with movement. Sensory and neural noises are known to induce stochastic resonance (SR), which improves information transfer to the subjects control systems, including the brain. As a consequence, sensory and neural noise provide behavioral benefits, such as stabilization of posture and enhancement of feeding efficiency. In contrast, the benefits of fluctuations in the movements of a biological system remain largely unclear. Here, we describe a novel type of noise-induced order (NIO) that is realized by actively exploiting the motion fluctuations of an embodied system. In particular, we describe the theoretical analysis of a feedback-controlled embodied agent system that has a geometric end-effector. Furthermore, through several numerical simulations we demonstrate that the ratio of successful reaches to goal positions and capture of moving targets are improved by the exploitation of motion fluctuations. We report that reaching success rate improvement (RSRI) is based on the interaction of the geometric size of an end-effector, the agents motion fluctuations, and the desired motion frequency. Therefore, RSRI is a geometrically induced SR-like phenomenon. We also report an interesting result obtained through numerical simulations indicating that the agents neural and motion noise must be optimized to match the prey's motion noise in order to maximize the capture rate. Our study provides a new understanding of body motion fluctuations, as they were found to be the active noise sources for a behavioral NIO.
Data Availability Statement: All relevant data are
within the paper.
Funding: This work was supported in part by the
New Energy and Industrial Technology
Development Organization (NEDO, http://www.
nedo.go.jp/english/index.html). The funders had no
role in study design, data collection and analysis,
decision to publish, or preparation of the
Competing interests: The authors have declared
that no competing interests exist.
The bodily movements of a biological system are noisy because of the stochastic nature of
biological sensory, neural, and actuation systems. Sensory and neural noise induce stochastic
resonance (SR), which provides a variety of benefits to an organism. These benefits include
improvements in information transmission to and through the neural system. Furthermore,
sensory and neural noise enhance cognitive performance [
], reflexes , feeding [4±6],
stochastic action selection [
], and memory-perception balance [
] (for other benefits of SR,
see the following reviews [10±12]). In contrast to the tremendous known benefits of neural
and sensory noise, the benefits of the noise inherent in motion fluctuations are largely unclear.
In fact, only a few SR-like benefits have been proposed for this type of noise (e.g.,
improvements in visual acuity due to eye tremor ).
The major reason that motion fluctuation is not considered to be a source of behavioral SR
or other noise-induced order (NIO) is that it exhibits a Lorentz-type spectrum and long-term
correlation. In fact, it has been shown that colored noise in general degrades the SR effect in a
nonlinear system [
]. Because the motion fluctuations of relatively large-bodied animals,
such as mammals, reptiles, and fish mostly have long correlation times, they cannot be used
directly as sources of noise to induce SR in a nonlinear system. However, these motion
fluctuations can still be helpful for a neurophysical agent. We find that by using the measure of
reaching success rate, we can observe a novel kind of NIO. Furthermore, we report that the bodily
motion fluctuations of a neurophysical agent provide aperiodic and stochastic input signals to
a feedback motion controller consisting of neurons. This leads to the emergence of neural
aperiodic SR [16±18].
In the following article, we report the results of our theoretical analysis of the reaching
success rate improvement (RSRI) of a Brownian particle that is controlled to reach a
periodicallymoving target. Next, we demonstrate that the RSRI ratio is dependent on the geometric size of
an end-effector used to catch a target. We further show the RSRI is a novel NIO based on the
mechanism of geometric-stochastic resonance (GSR), wherein the geometric size of the
endeffector, the frequency of the target movement, and the motion noise intensities interact with
each other and improve reaching success ratio. As an applicative and more general
experimental framework of GSR, we consider a numerically-simulated neurophysical agent with a
twodimensional body and a neural motion controller consisting of two arrays of
FitzHughNagumo neurons. Furthermore, we consider two experimental setups implemented using
numerical simulations: a static reaching task, wherein the agent tracks along a predesigned
path, and a dynamic capturing task, wherein the agent captures randomly moving objects.
Theoretical basis of RSRI via GSR
We consider an overdamped Brownian particle driven by a feedback controller as
x_ lx K
t0 x 2D0x
where = 0K/(λ + K), f = f0/(λ + K), D = D0/(λ + K). The probability density function of the
particle position P(x, t) is fully described by the following Fokker-Planck equation:
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Fig 1. Theoretical analysis of RSRI. (A) Schematic model of a feedback-controlled Brownian particle agent. The agent has an end-effector of size θ used to
reach a target moving along the pre-designed path xg(t). For simplicity, we assume that xg(t) is periodic. (B,C,D) Plot of theoretical hPRi with contour lines
versus the moving target frequency f and the agent motion noise intensity D computed using Eq (5) with θ = 0.01 (B), θ = 0.1 (C), and θ = 1 (D). (E) B with
respect to f and A = 0.1, 1, 2, 3 with = 1. Note that with t = 1/f, limf!1 B = A cos(1). (F) hPRi with respect to D × 10 and θ = 0.2, 0.4, 0.5, 0.55, 0.65, 0.8, 1 with
A = 0.1.
where g 1=
1 f 2, α = arccos(g). The analytical form for the reaching success rate PR(t),
which is the probability that the agent reaches the range of [xg(t) − θ, xg(t) + θ] is computed by
where s(t) = A cos(ft), A is the effective amplitude of xg, and is 0/(λ + K), B = c − gcα, c = cos
(ft), = 0 K(λ + K), cα = cos(ft + α). Note that the analytical form of the ensemble average of
PR(t) is identical to PR(t), that is, hPR(t)i = PR(t), where hzi denotes the ensemble average of z.
The peak PR(t) is calculated using D = Re(θB/log(±Q)), where Q
and Re() denotes taking the real part. This indicates that the optimal noise intensity needed to
maximize reaching probability is dependent on the interplay between the geometrical size of
the end-effector and the drive frequency.
The probability hPRi = hPR(t = 1/f)i exhibits two distinct modes based on the balance
between θ and B, as shown in Fig 1(F). Surprisingly, limD!0hPRi is limited to either 1 or 0, as
1; y > B
As an applicative and more general experimental framework for RSRI via GSR, we consider a
two-dimensional particle system driven by a nonlinear feedback controller consisting of two
arrays of FitzHugh-Nagumo neurons. The motion dynamics of the particle system are
where v is the velocity vector of the agent, f is the force generated by a neural motion
controller, γ is the friction coefficient and γ = 0.6 throughout this article, ξm(t) is a Gaussian noise
vector of unit intensity, and Dm is the motion-noise intensity.
The motion controller is designed to receive a feedback signal s(t) and outputs the force f(t)
based on the neuronal firing rate of two ensembles of FitzHugh-Nagumo (FHN) neurons, one
for each dimension, as
where K is the feedback gain, R0 is a offset variable, and R(t) is the firing rate of the neuron
ensemble. The dynamics of the ith FHN neuron of the jth ensemble is expressed as
The condition θ B corresponds to an unreachable regime, where the agent cannot reach the
target position without the help of noise, and hPRi is maximized by the optimal noise intensity,
as shown in Fig 1(B), 1(C) and 1(F). In contrast, θ > B corresponds to a reachable regime and
hPRi = 1 by D = 0, as shown in Fig 1(D) and 1(F). It may be counterintuitive that hPRi increases
following increases in f (i.e., a higher frequency leads to a more reachable condition) (Fig 1(B)
and 1(C)). This is because an increase in f results in a decrease in B, as shown in Fig 1(E), and θ
B leads to hPRi > 0.
where = 0.005, b is the bias signal, ξb is a Gaussian noise of unit intensity, Db is the bias
variability and Db = 0 unless otherwise stated, sj(t) is the input feedback signal to the jth neuron
ensemble, V is the fast variable, and W is the slow recovery variable. Independently of the
motion additive noise Dmξm(t), a neuron ensemble receives additive noise Dsξ(t) where ξ(t) is
the Gaussian noise of the unit standard deviation and Ds is the noise intensity. The firing event
t of a neuron is computed by
1; Vij > 0:5;
The mean firing rate of the jth ensemble is computed as Rj
t 1=N PiN1 Rji
t (N = 500
unless otherwise stated). The ensemble is in an excitable regime for b 0.274, and is in an
oscillatory regime (i.e., neurons spontaneously fire without any signal input) for 0.274 <
b < 0.3.
Fig 2. Neurophysical agent design and task setup. (A, B) Two different numerical simulation setups for studying behavioral NIO. In setup (A), we study the
NIO when a neurophysical agent tracks along a static predesigned path. In setup (B), we study the NIO that occurs when the agent captures randomly moving
(i.e., noisy) targets. In the second paradigm, we consider not only the additive neural and force noises internal to the subject agent, but also the motion noise
of the moving target.
The input signal s(t) encapsulates the computation performed by the neuron ensemble,
namely that of a positional PI (proportional-integral) feedback controller, as
t g e
where KI is gain for the error-integral control and g is the input gain. Furthermore, x(t) and
xg(t) are the current and desired agent positions, respectively. Note that xg(t) is predesigned in
the path-tracking experiment in Fig 2(A), and is dynamically updated at every simulation
time-step in the capturing experiment shown in Fig 2(B).
Numerical simulation of static reaching task
We consider two different kinds of tasks. First, we consider a static goal-reaching task. This
task consists of static path planning and path tracking. A static reaching task is the standard
motion control task for a biological system, e.g., arm-reaching or eye movement [20±22], and
for robot navigation. In this standard goal-reaching task, we assume that the target is fixed.
Agent motion fluctuations are influenced by the two noise sources, (1) the motion additive
noise Dmξm(t), and (2) the neural additive noise Dsξ(t).
In the numerical simulation shown in Fig 2(A), the agent is controlled to visit the four goal
positions (Xi, Yi) = (cos(1/4π + π/2i), sin(1/4π + π/2i)) (where i = 0, 1, . . ., 3) sequentially,
switching to a new goal every T [s] ((i = 0, 1, . . ., 3)). When the agent goal is switched to the
next one, the line from the agent's current position x(t = kT) to the next goal position t =
(k + 1)T is equally partitioned into T/Δt sub-goals xg. The agent is controlled to track this
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designed path xg(t). The offset R0 is computed as the time-average of R of the initial 100s in
every task simulation and this initial period is excluded from the performance analysis.
Performance measures of motion accuracy
Because the agent is scheduled to reach the kth goal at t = kT [s], we compute the distance de of
the agent position from the kth goal every T [s], and use de as a linear measure for the static
goal-reaching task. The ensemble average of de over different simulation trials, hdei is
1 Z Te
Furthermore, we use the average motion error hemi as another linear sensorimotor
where Te is a sufficiently long time. That is, hemi is computed by averaging the error across all
simulation time steps.
A nonlinear performance measure: Goal-reaching success rate
In addition to the measures de and em, we use a measure that is obtained by applying a
nonlinear function to de. A straightforward example of such a nonlinear measure involving using a
threshold function to digitize the distance de is
where de is the distance of the agent from the goal position. Note that the measure SR(de) is
applicable to many biological tasks, such as capturing prey, and to reaching tasks, where a
system's physical body is required to be within a certain range of an object within a certain time
period. We use the ensemble average of the goal-reaching success rate hPRi as a task evaluation
de is calculated every at t = kT [s], and the ensemble average of PR, hPRi is
1; de < y;
Numerical simulation of dynamic capturing task
In the numerical simulation shown in Fig 2(B), we consider a dynamic reaching task where
the goal position (i.e., the position of the target objects) moves. In this setup, we use Brownian
particles as target objects. Therefore, the motion of the target object provides an additional
source of agent motion fluctuation. Note that in the framework (B), there is no pre-designed
path to track and the agent goal position is updated at every time step based on the movement
of the target object.
In the simulation environment, Np moving target objects are located randomly within
range [−L, L] (we use Np = 100 and L = 7.5 in this paper), as shown in Fig 2. The ith target
object moves randomly based on the dynamics
where mg = 0.1, vg,i is the velocity, zi(t) is the Gaussian noise of unit variance, and Dp is the
noise intensity. The target object is subjected to a force Fw from a virtual wall when it moves
beyond the region [−L, L].
The agent is controlled to pursue the position of a moving target. That is, xg(t) = xp(t)+vp(t)
Δt, where xp(t) and vp(t) are the position and the velocity of the current moving target,
respectively. When the distance dp between the agent and the target satisfies dp < θ, the moving target
is ªcapturedº and is removed from the simulation. After the agent captures a certain target, the
target is switched to the nearest moving object. The offset R0 is computed as the time-average
of R of the initial 300s in every task simulation and this initial period is excluded from the
The capturing rate per unit time is Cr, which provides a nonlinear measure of the agent's
sensorimotor performance on this task. This is computed as
1 Z T
where Θ(z) is the Heaviside step function.
Benefits of noise in the behavior of a neurophysical agent
For Ds = 0 and Dm = 0, the system exhibits fully deterministic behavior, although it may exhibit
jittering due to the overshooting characteristics of a simple feedback controller. In this
deterministic regime, the neural system exhibits a totally synchronized firing pattern across
different initial neuronal conditions (see Fig 3 for both the motion-control signal [S1, S2] and the
neural firing-rate time series [R1, R2]). Clearly, in this regime, neural spike frequency encodes
the motion control signal. The agent-movement and motion-control signals become stochastic
and aperiodic with either Dm > 0 or Ds > 0. For relatively large Ds values, the neuronal firing
rate also becomes asynchronous. Note that the combination Dm > 0 and Ds = 0 can generate
an aperiodic motion-control signal, but it cannot generate an asynchronous neuronal firing
Motion accuracy improvement by SR in a neural motion controller. Fig 4 shows that
the best motion accuracy, i.e., the minimum hdei or the minimum hemi, is realized when there
is nonzero neural noise Ds [Fig 4(A)±4(C)]. Interestingly, the peak positions for hdei and hemi
are very different (i.e., the minimum hdei is calculated using Ds 0.01, and the minimum hemi
is calculated using Ds 0.06).
The measure hemi has a strong dependency on the neural performance measures ρ, the
correlation coefficient of the input control signal and the neuronal firing rate, and No, the
timeaveraged product of the s(t) norm and R(t) norm. Here, ρ and No are computed as r
tdt=No and No 1=Te 0Te
t kk R
t k dt.
We can see that there exist two kinds of SR-based motion accuracy improvements: a
subthreshold SR corresponding to a small input gain g [Fig 4(E)], and a suprathreshold SR
corresponding to a large input gain g [Fig 4(F)]. With the small input gain g * 0, the agent motion
tends to overshoot the desired path due to the poor information transmission to the motion
controller [Fig 4(E), bold gray line for Ds = 0 and solid red line with Ds = 0.06]. In contrast,
with the large input gain g 0, the agent motion tends to oscillate around the desired path
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Fig 3. Emergent aperiodic control signal and asynchronous neural firing. (S1, R1) The input signal to the motion actuator (S1) and the corresponding
neural firing rate R(t) − R0(R1), with Ds = 0 and Dm = 0. Note that the input signal to the actuator is totally deterministic, although it exhibits jittering. In addition,
the corresponding neural spikes are synchronized (the even vertical lines represent bursts of spikes, not individual spikes.) (S2, R2) An aperiodic and
stochastic control signal emerges with either Dm > 0 or Ds > 0 (S2). The corresponding firing rate becomes asynchronous if Ds > 0 (R2).
due to the hard synchronization of neuronal firing [Fig 4(F), bold gray line for Ds = 0]. This
synchronized neuronal firing is due to the poor pooling ability of the motor controlling
neurons (the panel (F) is obtained using Db = 0.1). The large bias variability Db 0 can improve
the pooling ability in the noiseless neural controller Ds = 0. However, it must be noted that Db
0 obscures the SR effect for Ds > 0. Furthermore, the motion accuracy provided by SR is
higher than the motion accuracy realized using noiseless pooling with Db 0 and Ds = 0, as
shown in panel (G). Note that g 0 is exactly the case wherein SR growth [
Furthermore, we could not find any motion accuracy improvements due to the presence of force
noise (Fig 5).
Improvement in static reaching success rate
The distance de, which was used in the previous study, represents the ªlinearº difference
between the agent and the position of the goal. Because this difference is linear, de exhibits a
monotonic increase in response to the intensity of the additive force noises. This implies that if
we use a certain measure with a nonlinear dependence on the agent and goal positions, we
may observe benefits of motion noise in the sensorimotor task. In fact, as expected from the
theory of GSR, we observe the benefits of the noise when we use reaching success rate as a
measure, as shown in Fig 6.
Fig 6 shows the goal-reaching success rate as a function of the neural and motion additive
noises Ds and Dm in the experimental setup shown in Fig 2(A). With the parameter sets shown
in Fig 6(A), 6(B), 6(D) and 6(E), the agent cannot achieve a good reaching success rate using
the default deterministic feedback control because force feedback gain K is not sufficient. In
this ªdeterministically unreachableº parameter region, the combination of nonzero neural
noise and nonzero motion noise leads to an improvement in the goal-reaching success rate.
This realization of RSRI can be interpreted as a result of the interplay between the motion
noise and the neural noise: motion noise generates an aperiodic input signal in the neural
system, as shown in Fig 3(S1), 3(R1), 3(S2) and 3(R2), and neural noise generates an aperiodic
neural SR [16±18]. Note: in the deterministically reachable region in Fig 6(C) and 6(F) (i.e., PR
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Fig 4. Motion change due to the presence of neural and force noises. (A, B) hdei and hemi with the parameters T = 10, K = 10, g = 0.02. (C) hdei with Dm
= 0.01 and hemi, with Dm = 0.05 as a function of Ds. (D) hρi and hNoi as a function of Ds. (E, F) The change in motion trajectory due to the presence of neural
and motion noises, with T = 10, K = 10, and g 1 [(E)] and g 0 [(F)]. The inset is an enlargement of the respective areas inside the rectangles. Note that
the bias variability Db 0 leads to high pooling ability and reduces the oscillatory motion, but obscures the neuronal SR effect. Furthermore, the motion
accuracy achieved due to neuronal SR (with Db * 0 and Ds > 0) is higher than it is in the noiseless system with high motor pooling ability (with Db 0 and Ds
= 0) (G). Numerical hdei, hemi, hρi, and hNoi are computed from 500 trials of a 500 s numerical simulation.
> 0 for Ds = 0 and Dm = 0), the PR improvement effect due to motion noise is in principle
unobservable (because PR = 1 by default).
Improvement in the capturing rate due to environment-agent noise balancing
In the dynamic reaching experiment, we consider a task where the agent is controlled to
capture moving prey. Furthermore, we investigate how the capture number per unit of time
depends on the motion fluctuations of the agent and those of the moving targets.
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Fig 5. Motion error with respect to Ds. hemi with g = 0.02, Db = 0.1, and Ds = 0.05 for (A), and g = 0.5, Db = 0.1, and Ds = 0.005 for (B). The error bars
indicate standard deviations. Note that we could not find any significant improvements due to the presence of force noise Dm. Numerical hemi is computed
from 500 trials of a 500 s numerical simulation, and the error bars correspond to the standard error.
The dependence of capture rate hCri on Dm and Dp is shown in Fig 7(A) and 7(B), with
K = 0.5 and θ = 1. Clearly, hCri is improved with the presence of motion additive noise Dm.
Interestingly, hCri is also improved with the presence of Dp, which indicates the prey's motion
noise. Furthermore, hCri is a function of Dp, Dm, and Ds. These interesting results imply that
the ability to capture is a function not only of agent motion and neural noise, but also of the
prey's motion noise. From the point of view of the capturing agent, the neural and motion
noises must be adjusted to match the intensity of the prey's motion noise. On the other hand,
for the prey to avoid being captured, it must adjust the intensity of its motion noise away from
that of the agents. This experimental result implies that biological systems in the context of
survival competition will control their neural and motion noise intensities based on their
environmental noise levels.
The improvement in the capturing rate hCri due to Dm is dependent on the size of the
geometric threshold θ, as shown in Fig 8. It may be reasonable to presume that a larger biological
agent can more efficiently exploit the GSR that is induced by motion fluctuations when
capturing small targets.
Ref. [4±6] report that the feeding rate of paddlefish (capturing rate of planktons per minute)
is improved in the presence of electrical sensory noise. Conventionally, this feeding behavior
improvement has been thought to be the result of sensitivity improvement due to the presence
of electrical sensory noise. Our results may imply that, in addition, the electrical signal noise
induces motion fluctuations in the capturing agent, which then help to improve the capture
The effect of motion noise on the capturing task is summarized as follows: motion noise
enables reaching that is not obtainable deterministically, and the optimal motion noise
intensity is determined by a balance with the target motion noise.
Dynamic capturing rate improvement using a simple PI feedback controller
To determine whether our results have general applicability, we obtained experimental data
using a simple PI motion controller that did not have any neurons. The force output of this
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Fig 6. The goal-reaching success rate hPRi as a function of motion noise Dm and neural noise Ds. The task parameters are T = 10, K = 0.5 − 1.5 in
panels (A±C), and T = 5, K = 2 − 5 in panels (D±F). Additive motion noise improves reaching success rate when K is not sufficient to produce a 100%
goalreaching success rate (this is shown in panels (A), (B), (D), and (E)). As shown in panels (C) and (F), if K is large enough to realize a 100% success rate, the
PR monotonically decreases with Dm. Numerical hPRi are computed from 100 trials of a 500 s numerical simulation with θ = 0.1 and N = 100, and the error bars
correspond to the standard error.
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simple PI motion controller is described as
t K e
where e(t) = xg(t) − x.
Although several parameter adjustments are required, it is possible to reproduce the results
of Figs 7 and 8. Fig 9(A) shows the improvement in reaching success rate, and Fig 9(B1)±9(B3)
Fig 7. Improvement in capture rate hCri due to motion noise Dm and the noise of the targets motion Dp. The ensemble average of the capture rate hCri
as a function of Dp and Dm with internal neural noise Ds = 1 × 10−3 (A) and Ds = 5 × 10−3 (B). The other parameters are K = 5, g = 10−2, b = 0.24, N = 100, and
θ = 1. The peak hCri is distributed roughly along the line Dp + Dm = 0.4 in (A) and Dp + Dm = 0.3 in (B). It is clear that the maximization of hCri requires a
balance among Dm, Ds, and Dp. Numerical hCri are computed from 400 trials of a numerical simulation.
Fig 8. Capture rate modification by threshold size θ. The parameters are Ds = 1 × 10−3, K = 5, b = 0.24,
N = 100, and g = 0.01. The rate of improvement in capture rate is dependent on the size of the geometric
threshold θ. Numerical hCri are computed from 100 trials of a 500 s numerical simulation, and error bars
indicate standard errors and are within the symbols.
show the improvement in the capture rate. These results support the idea that behavioral SR in
reaching and capturing tasks is a general property of feedback-controlled physical agents.
We investigated NIO in the context of sensorimotor coordination in a neurophysical 2D
particle agent. The motion controller of the agent consisted of an FHN neuron ensemble. The
addition of neural noise to the controller led to an improvement in the agent's motion accuracy, as
shown in Fig 4(A)±4(C). The motion accuracy improvement by the addition of neural noise
would be primarily a consequence of the neural SR that optimizes the controller feedback
output, i.e., the maximization of ρ as shown in Fig 4(D). It must be noted that the addition of
neural noise decreases the variance of a neural output [i.e., hNoi shown in Fig 4(D) decreases by
the addition of neural noise]. Because the addition of force noise to the agent body
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Fig 9. Behavioral SR of an agent driven by a simple non-neural PI controller. (A) Improvement in the goal-reaching success rate due to additive motion
noise. Numerical hPRi are computed from 40 trials of a 500 s numerical simulation with KI = 0.01 and θ = 0.1. Error bars in (A) indicate standard deviations.
(B1±B3) Capture-rate improvement due to motion additive noise. Numerical hCri are computed from 1,000 trials. The parameters for (B1±B3) are KI =
0.02 × 10−2 and θ = 2.
monotonically degraded motion accuracy, as shown in Fig 5, motion accuracy improvement
by the addition of neural noise would be a secondary consequence of the decrease in the neural
Although motion fluctuations per se degrade motion accuracy, we found that a nonlinear
performance measure such as goal-reaching success rate can exhibit the emergence of an NIO
induced by the the motion fluctuations. Particularly, we found that motion fluctuations
improved the goal-reaching success rate hPRi, as shown in Figs 6±9.
Interestingly, for a neurophysical agent hPRi was a function of force additive noise intensity
Dm and neural additive noise Ds as shown in Fig 6. Furthermore, in a capturing task where not
only the neurophysical agent but also the prey's motion was noisy, the capturing success rate
hCri was a function of agent force noise intensity Dm, neural noise intensity Ds, and prey's
motion noise intensity Dp as shown in Figs 7 and 9. These results imply that biological systems
may handle the balancing of motion and neural noise dependently on the environmental
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Task success ratio as a marker of NIO
It should be noted that we did not find any benefit of motion fluctuation based on measures of
the relationship between predesigned paths and the actual motion trajectories. Likewise, we
could not find SR-like benefits using measures of the distances among the goal positions and
the final agent positions. Only when using measures of the discretized state probability, such
as goal-reaching success rate and capturing rate, did we observe SR phenomena induced by
It is worth noting that most of the conventional systems for studying SR require a
discretized dynamical representation or a digitized output. These systems include traditional
double-well potential systems, threshold systems, the spiking neuron, and two-state dynamical
systems (e.g., the FitzHugh-Nagumo neuron). These systems are capable of digitizing the
input signals or generating digitized output (for reviews of the conventional SR-capable
systems and frameworks, see Refs. [11, 23±26]).
Furthermore, the conventional framework for detecting a weak signal requires a discretized
measure (i.e., detection or no-detection). In behavioral frameworks, responding to weak and
subtle sensory signals also requires a digitized response (i.e., response or no-response). A
recent concept for studying SR, called entropic SR [
] and GSR [
], posits that Brownian
particles move between two rooms connected by a narrow aperture. In this case, the setup of
the two rooms provides the state digitization. (Note that the mechanism of GSR described in
Eq (5) would be very different from the conventional mechanism of GSR reported in Refs. [
], although both studies share the common characteristic that the interaction of noise and
geometric constraints induce NIO). In this manner, a digitized measure, or digitizing
dynamics, may be implicit requisites for observation of the SR. At present, this idea is only an
inference from analogy and requires theoretical analysis.
Measures of discretized-state probabilities, such as the goal-reaching success rate and
capturing rate can be generalized as task success ratios. It should be noted that the measure ªtask
success ratioº is a highly nonlinear function of a variety of arguments, such as the
appropriateness of feedback gain, neural noise intensity, neuron size, input signal gain, input-output
information, and the distance from the goal position. Therefore, the task success rate may be able to
implicitly represent the extent to which calculations of these arguments are improved by some
In this paper, we investigated the prospective benefits of the bodily motion fluctuations of an
embodied physical agent. We considered a static path-tracking task and a dynamic capturing
task for moving prey agents. We found that motion fluctuations degrade motion accuracy, but
improve the reaching success rate and capturing rate. These results imply that a biological
agent may exploit bodily motion fluctuation in several behavioral tasks, such as reaching,
capturing, and navigation, by adjusting the intensity of the motion noise.
We are grateful to Prof. Yoshiharu Yamamoto (Univ. of Tokyo) for much valuable discussion.
This research is based on results obtained from a project commissioned by the New Energy
and Industrial Technology Development Organization (NEDO).
Conceptualization: Shogo Yonekura.
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Data curation: Shogo Yonekura.
Formal analysis: Shogo Yonekura.
Funding acquisition: Yasuo Kuniyoshi.
Investigation: Shogo Yonekura.
Methodology: Shogo Yonekura, Yasuo Kuniyoshi.
Project administration: Yasuo Kuniyoshi.
Resources: Yasuo Kuniyoshi.
Software: Shogo Yonekura.
Supervision: Yasuo Kuniyoshi.
Validation: Shogo Yonekura, Yasuo Kuniyoshi.
Visualization: Shogo Yonekura.
Writing ± original draft: Shogo Yonekura.
Writing ± review & editing: Yasuo Kuniyoshi.
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