Noisy Adaptation Generates Lévy Flights in Attractor Neural Networks
Authors: Xingsi Dong
Tianhao Chu
Tiejun Huang
Zilong Ji
Si Wu
Venue: NeurIPS 2021 Poster (2021)
Keywords: Lévy flights
Neural adaptation
Attractor neural networks
Information searching
Free memory retrieval

Abstract: Lévy flights describe a special class of random walks whose step sizes satisfy a power-law tailed distribution. As being an efficient
searching strategy in unknown environments, Lévy flights are widely observed in animal foraging behaviors. Recent studies further showed that human cognitive functions also exhibit the characteristics of Lévy flights.  Despite being a general phenomenon, the neural mechanism at the circuit level for generating Lévy flights remains unresolved. Here, we investigate how Lévy flights can be achieved in attractor neural networks. To elucidate the underlying mechanism clearly, we first study continuous attractor neural networks (CANNs), and find that noisy neural adaptation, exemplified by spike frequency adaptation (SFA) in this work,  can generate Lévy flights representing transitions of the network state in the attractor space. Specifically, the strength of SFA defines a travelling wave boundary, below which the network state displays local Brownian motion, and above which the network state displays long-jump motion. Noises in neural adaptation causes the network state to intermittently switch between these two motion modes, manifesting the characteristics of Lévy flights. We further extend the study to a general attractor neural network, and demonstrate that our model can explain the Lévy-flight phenomenon observed during free memory retrieval of humans. We hope that this study will give us insight into understanding the neural mechanism for optimal information processing in the brain.

Noisy Adaptation Generates Lévy Flights in Attractor Neural Networks
Xingsi Dongl.2.3: Tianhao 12,,,, Tiejun Huang', Zilong Ji Si Wu2.2,31
1, School of Psychology and Cognitive Sciences, Peking Univerisity.
2, IDG/McGovern Institute for Brain Research, Peking University.
3, PKU-Tsinghua Center for Life Sciences,
Academy for Advanced Interdisciplinary Studies, Peking University.
4, School of Electronics Engineering and Computer Science, Peking University.
Equal contributions. f; Corresponding authors.
{dxs 9980609 chutianhao, tjhuang, j pizilzloe long, www.uupku..c.mmm edu. cn
Abstract
Lévy flights describe special class of random walks whose step sizes satisfy a power-law tailed distribution. As being an efficient searching strategy in unknown environments, Lévy flights widely observed in animal foraging behaviors. Recent studies further showed that human cognitive functions also exhibit the characteristics of Lévy flights. Despite being a general phenomenon, the neural mechanism at the circuit level for generating Lévy flights remains largely unclear. Here, we investigate how Lévy flights can achieved in attractor neural networks. To elucidate the underlying mechanism clearly, we first study continuous attractor neural networks (CANNs), and find that noisy neural adaptation, exemplified by spike frequency adaptation (SFA) in this work, can generate Lévy flights represent- ing transitions of the network state in the attractor space. Specifically, the strength of SFA defines travelling wave boundary, below which the network state displays local Brownian motion, and above which the network state displays long-jump motion. Noises in neural adaptation cause the network state to intermittently switch between these two motion modes, manifesting the characteristics of Lévy flights. We further extend the study to general attractor neural network, and demon- strate that our model can explain the Lévy-flight phenomenon observed during free memory retrieval of humans. We hope that this study will give us insight into understanding the neural mechanism for optimal information processing the brain.
Introduction
Lévy flights, also termed anomalous diffusion or super diffusion, refer special class of random walks whose step sizes follow distribution with long power-law tail. Mathematically, this power-law tailed distribution expressed as
p(x) 1-o
(1)
where the step size x can be measured either in the spatial domain, e.g., the length of movement, or in the temporal domain, e.g., the time interval between successive events. The parameter a is called the Lévy exponent satisfying < a 2. When 2 2, the above process degenerates to Brownian motion due to the Central Limit Theorem [1] Compared to Brownian motion, whose step sizes satisfy Gaussian distribution, Lévy flights are much more likely to generate large step sizes. Thus, it is often intuitively stated that Lévy flights are composed of frequent local motion (similar to the Brownian motion) and intermittent long-jump motion (Fig. In contrast Brownian motion
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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which tends to oversample local area, Lévy flights with long jumps provide a more efficient way to search for scarce targets that are randomly distributed in an unknown environment, and the highest search efficiency occurs at a 2 3
(C)
(D)
n
Lavy ligha Brownion motion
Simuston on mn
Lovy ghg
Browrean medton
10
02 04 D6 03 10 12
Levy flights
Brownian motion
Step size
logistep size]
Figure I: Illustrations of Lévy flights and Brownian motion. (A) An example of Lévy flights generated by sampling step sizes from Eq. M. with a chosen between 0 and 2. (B) An example of Brownian motion generated by sampling step sizes from Eq. (T). with a larger than 2. Both Lévy flights and Brownian motion showed here are generated with the same simulation duration, i.e., 200 time steps. (C) The histograms of step sizes (æ in Eq. ① of Lévy flights (blue) and Brownian motion (orange), respectively. Notably, Lévy flights have ultra-long jumps which are absent in Brownian motion. (D) The log-log plot of the histograms of step sizes in (C) shows that the distribution of Lévy flights follows power-law tail.
As being efficient strategy for information search, Lévy flights have been widely observed in for- aging behaviors across different animal species, ranging from microzooplankton [5] to drosophila [6] and from albatrosses 7 to spider monkeys [8] Similar foraging-like patterns also observed in human behaviors, e.g., individual mobility in the geographical scale (). Experimental studies further revealed that Lévy flights also exist in human cognitive functions. For instance, has been found that the long tailed distribution holds for the saccadic eye movement in random quenched salience field [10), for the time interval between successive retrieved items in free memory retrieval task (TI) and for the mental exploration of 'bid space' in Lowest Unique Bid Auctions (LUBA) [12]. Recently, an fMRI study unveiled that the dynamics of neural activation at the resting state of the brain can also be characterized by Lévy flights [13. In a more detailed study, Pfeiffer and Foster showed that the awake reply trajectories of the hippocampal place cell ensemble in immobile rats resemble superdiffusion dynamics [1),, which is very similar to Lévy flights. Intriguingly, Lévy flights have also attracted considerable attention in the deep learning community for modeling the optimization process with the stochastic gradient descent (SGD) [5 [7]
Despite that Lévy flights have been widely observed and that various computational models have been proposed to account for their occurrences in different scenarios (see, e.g., [8), the neural basis for the brain generating Lévy flights remains far from resolved (see more discussions in Sec. 4. In the present study, we will investigate the neural mechanism for generating Lévy flights at the circuit level, i.c., how a neural network with appropriate biological features generates Lévy flights in the space of information presentation. We hope that this study will help us to further understand the emergence of Lévy flights in cognitive functions of humans.
Specifically, we will study attractor neural networks which model how information is represented in neural systems [19 20),, and hence Lévy flights in the corresponding attractor spaces reflect information processing in the brain. To elucidate the underlying mechanism clearly, we focus on studying continuous attractor neural networks (CANNs), as this allows dynamics analytically. The key characteristic of Lévy flights is the occurrence of intermittent long jumps. In order to achieve this property, we consider noisy adaptation in the neural dynamics. which is exemplified by spike frequency adaptation (SFA) in this work. The role of adaptation is to induce a spontaneously moving state, called travelling wave, in the network dynamics, which occurs when the strength of SFA exceeds a threshold. Thus, if the mean SFA strength is set slightly below the threshold, noises in adaptation will occasionally push the SFA strength to exceed the threshold. Consequently, the network state will temporarily fall into the travelling wave state and experience long-jump movement in the attractor space. Over time, noisy adaptation causes the network state to switch intermittently between local motion (when the SFA strength remains below the threshold) and long-jump motion in the attractor space, manifesting the characteristic of Lévy flights. We carry out theoretical analyses to formally verify the above mechanism, and simulation results agree with
solve the network

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our theoretical analyses very well. We further extend the study to general Hopfield-lik attractor network, and use it to model the Lévy-flight phenomenon as observed in the free memory retrieval experiment [I]]. We hope that this study will help us to understand how neural circuits generate Lévy flights, and give us insight into understanding the emergence and the computational roles of Lévy flights in brain functions.
2 Lévy flights in continuous attractor neural networks
We first study the generation of Lévy flights in continuous attractor neural networks (CANNs). CANNs have been widely used as canonical models for elucidating the encoding of continuous features in neural systems 21 22
23 24 including, for instance, the orientation [25], the head
direction [26], and the spatial location 27
28.. The advantage of studying CANNs is that it allows [29] and hence gives insight into understanding
us solve the network dynamics analytically
the computational mechanism of generating Lévy flights. In Sec. 3 we will extend the theoretical findings to other attractor networks.
2.1 The model
(A)
83
(D)
02
f)
Neuronal connactions
V(x. O
Moving direction
J(sx))
Self-relaxation
Recurrent interaction
Spike frequency adaptation
(E)
x10
neuronal sheet
External noise input
Intrinsic speed [sae
022
Energy
Network response
Travelling wave boundary
Attractor space
0.5
1.
29
Scaled SFA strength m/ (f/%)
Figure 2: A two-dimensional (2D) continuous attractor neural network (CANN) with neural adapta- tion. (A) Illustrating the network structure. Top: the excitatory connections between the neuron and neurons at other locations. Middle: the two-dimensional neuronal sheet on which neurons are uniformly distributed. Bottom: the network response with Gaussian bell shape. (B) The kinetics of a single neuron, determined by the self-relaxation, recurrent interaction, spike frequency adaptation, and external input. (C) the attractor space formed by all stationary states of the network, on which the CANN is neutrally stable (for clearance, the case of 1D is shown). (D) The network state with SFA as a negative feedback in the 2-D CANN feature space. The bump (xx t) can move spontaneously in the attractor space in any direction depending the initial state. The bump of the adaptation current V(x, () lags behind U U(x, () with a separation s(t) along the moving direction, due to the delayed feedback modulation of SFA. (E) The intrinsic speed of the travelling wave ||Vinell vs. the SFA strength (scaled by 7/ .T.).. The red point represents the travelling wave boundary with m T/T.-
As illustrated in Fig. A we consider two-dimensional CANN, in which neurons are uniformly distributed on rectangular neuronal sheet according to their preferred feature values. Denote U(x, t) as the synaptic input to the neuron x, with x = (æ11 .22) and æ 22 E (-o0, oc), and r(x, t) the corresponding firing rate. The dynamics of Uxx, t) is determined by its own relaxation, the recurrent inputs from other neurons, the neural adaptation, and the external noise input (Fig. B which is written as,
OU (x, (x,1) Øt
(x, - / J xx,x'') (x', dx' - V (x, + outu (x,t).
(2)
Here, 7 the synaptic time constant and p the neuronal density. The term ay (x, 1) represents the external noise input, with oy the noise strength and &u (x, t) the Gaussian white noise of zero
3

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mean and unit variance. In this study, we propose an internal mechanism to generate Lévy flights, therefore we only consider noisy imputs of zero mean to the network and do not consider non-zero drift inputs (see Sec. 4 for more discussions). The recurrent neuronal connections, J(x, x') = with ||x- - x = (21 - 21)² (22 z2)², are translation- invariant on the neuronal sheet (Fig. 2 which means that [xx,x') is a function of ||x - |x'|- The nonlinear relationship between the firing rate r(x; () and the synaptic input U(x, 1) is implemented by divisive normalization, which is written as,
(o/(2ma)) |-||x x'1l
(x t) r (x, t) 1 kp Sx U2 (x',
(3)
Here, the parameter k controls the normalization strength. In reality, divisive normalization could be implemented by shunting inhibition (30].
The term V (x, t) on the right-hand side of Eq. 2 represents an adaptive current. Adaptation is a general phenomenon referring to that a neuron population generates negative feedback to suppress its response when the activity level is high. Neural adaptation can result from different mechanisms, and their effects on the network dynamies are similar (see Sec. for more discussions). Here, we consider spike frequency adaptation (SFA) at the single neuron level as an example. In reality, SFA could be implemented by the interplay between calcium currents and intracellular calcium dynamics with calcium-gated potassium channels [31] The dynamics of Vxx, t) is written as,
av (x, t) -V (x, + [m + Jm (x, t) U (x,t), a
(4)
where Tu is the time constant of SFA, with To >> T, indicating that SFA is process much slower than neuronal firing. m is the mean SFA strength, with om the noise strength and (m t) denoting
the Gaussian white noise of zero mean and unit variance.
We first review the properties of the CANN without adaptation, i.e., by setting m 0 and o. = in Eq. { It has been shown that without the external noise input (ay 0), the CANN holds a continuous family of Gaussian -rassian-shaped shaped stationary states, called bumps. when the global inhibition amplitude & is smaller than a critical value kic = pJ8 220/(335 (322ma))
(29]. These bump states are expressed where z is a free parameter representing the bump position, i.e., a feature value encoded by the network, and Ay is a constant representing the bump height. These bump states form attractor space (Fig. (
as (x) = Au exp |- 2ll//(4a))
which the CANN is neutrally stable, which means that a small external input can drive the bump to move smoothly in the attractor space without distorting its shape. This neutral stability is the key that enables CANNs to realize accurate path integration [27] 28). smooth tracking of a moving object [32, and efficient population decoding (33] Specifically, under the drive of external Gaussian noises, the bump will exhibit Brownian motion in the attractor space [34].
We further review the properties of the CANN with a fixed adaptation strength (i.e., m is a constant and om = 0). It has been shown that without external input, the CANN holds a moving bump as its stationary state when m > T/Tu, which is called travelling wave (Fig. 2 33 This state reflects the intrinsic mobility of the network, as the bump moves spontaneously in the attractor space without relying on an external drive. The travelling wave state is expressed as t(x,t)
V Au exp [- - vinat) /?/(4)))) with || = 2a/Te / ///// /mTe/T the travelling speed and A'r the bump height B51 The mechanism underlying this intrinsic mobility is intuitively understandable. Suppose that the bump initially appears at an arbitrary location in the attractor space. Due to SFA, those most active neurons, i.e., those around the peak of the bump, receive the strongest adaptation current, and their activities are suppressed the most. Since those neighhoring neurons are less active and hence are less suppressed, the bump tends to move away to the neighborhood due to recurrent connections and competitions via divisive normalization; after moving to the new location, the suppression and competition start again. As a result, the bump will keep moving in the attractor space. Analogous to the situation without adaptation, if the SFA strength is fixed and m < T/Te, external noises will only induce Brownian motion of the bump in the attractor space.
2.2 Noisy adaptation generates Lévy flights
Before going to the detailed mathematical analyses, we first provide an intuitive understanding of how noisy adaptation leads to Lévy flights. As introduced above, CANN with SEA has the intrinsic

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mobility of generating a travelling wave when the SFA strength is sufficiently strong. The condition of m = T/T, defines the travelling wave boundary. Above the boundary, the bump will move spontaneously in the attractor space; below the boundary. the bump will either remain static if no external input exists, or exhibit Brownian motion when external noises are applied. The interesting phenomenon emerges if the SFA strength is noisy and its mean value m. is close to the boundary. In such case, fluctuations (due to the noise term ( t) in Eq. (4) will push the SFA strength to cross the boundary occasionally, which causes the network to fall into the travelling wave state temporarily, and consequently, the bump travels over a long distance (a long jump) in the attractor space. Over time, along with fluctuations of the SFA strength, the bump displays intermittent local motion (when the SFA strength remains below the threshold) and long-jump motion, manifesting the characteristic of Lévy flights. We present the formal analyses below.
Consider that the noise strengths in Eqs. 2 are sufficiently small, such that their effects on distorting the bump shape are negligible, and we assume that the network state has the Gaussian form as in the static case, which given by,
[x zxt)]
U tx),) = At(t) exp
(5)
4a²
[x - z(t)1²
r(x,1) = exp
(6)
2a
[x (a(1) - -(()))
V(x,t) = oxp
4a
where (t). (t) and (t) represent the heights of bumps Uxx, t), r(x, f) and Vxx, f), respec- tively. z(t) is the center of bumps U(x, t) and r(x, t), whose trajectory reflects changes of the network state in the attractor space. z(t) s(t) is the center of bump V(x, t). with s(t) denoting the separation between Uxx t) and V(x, t). Note that the bump V(x, t) always lags behind U(x. t), reflecting that the adaptation current is delayed with respect the neural response (Fig. 2D)
Previous works have shown that the dynamics of a CANN is dominated by very few motion modes 29 36). Therefore, to solve the network dynamics, we can project the network dynamics onto those dominating modes and simplify the analyses significantly" In the current study, we adopt the first two dominating motion modes, corresponding to the changes of bump height and position, respectively, which are given by,
exp {
[x - z(t)
ui (x|z)
(8)
2m
4a²
{
[x
"1 (x|z)
[x z(t) exp
a² 2T
4a
By projecting the network dynamics onto these two modes, we obtain the dynamics of the bump height and position, and the latter reflects how the network state changes in the attractor space. Specifically, by substituting the presumptive network state Eqs. (5 into the network dynamics Eqs. 2 and then projecting them on the motion mode of bump height Eq. (8) we obtain the dynamics of bump heights (see SI.1 for the details), which are written as,
dAa
Aa - Ag + JopA, =
OU Ev,o(t),
+
(10)
dt
2 av2T
dA., -A, mA +
0 Av Am
To di
(1),
(11)
20
where (t)o(t) and (m,a(t) denote, respectively, the projected noises of Exr and &m (1) on the bump height mode, which are still Gaussian white noises of zero mean and unit variance. Note that do not explicitly write down the dynamics of A,, as it has deterministic relationship with A., that is, we obtain A, A (1 + 2Ta kp
by substituting Eqs. 6 into Eq. 3
Similarly, substituting the network state Eqs. 7 into the network dynamics Eqs. 4 and then projecting them on the motion mode for bump position Eq. (9), we obtain the dynamics of bump
Projecting a function fx) on mode u(x) means computing fx f(x)u(x))d)
5

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positions (see S1.1 for the details), which are,
T dt
dz
OU /2 Au 8 +
Av
Au V (0,
(12)
ds
Au mAy AmAuEm, 2A.VTa
ToJU 2 A,,
AmAu A
To
dt
TAu
A,
+
(v,
2T
S,,
(13)
Here ( and (t) denote, respectively, the projected noises of (t) and Em (t) on the bump position mode, which are also Gaussian white noises of zero mean and unit variance.
We can obtain the stationary distributions of A, and A, in Eqs. (T0% by solving the corresponding Fokker-PPannce equations, which is a general method to describe the time evolution of the probabilistic density function of a moving particle in physics. Moreover, since oy and om are very small, the variances of Au and A, can be ignored compared to their mean values. Under this approximation, can be further simplified by replacing A, and A, with their mean values A, and Ag and using the approximation A, = mA, according to Eq. 111. which are written as,
Eqs. 11 13)
dz
+ aU 2 A, V (t),
ms
(14)
To dt
ds
1 m
2 7am (m.(tt)
m
s
2
ToOV T Au
+
& (t).
(15)
2m
m
Here (1) is a newly defined Gaussian white noise of zero mean and unit variance (by combining
the last two noise terms in Eq. (3). Eq. (14) shows that the bump position z(l) is determined by
drift term reflecting the contribution of SFA and a diffusion term reflecting the contribution of the
external noise input. Apparently, when no adaptation exists, i.e., no drift in Eq. (14) (m 0), the bump movement is Brownian motion due to the diffusion noise.
To solve the dynamics of the bump position z(t), it is necessary to first solve the dynamics of s(t) expressed in Eq. (15). We see that the drift coefficient of s(t) in Eq. (15) consists two parts. The first part, 1 mT./T, T, measures the distance of the mean SEA strength to the travelling wave boundary (given by T/T,),, and we denote μ = mT, T as the to-runccatt--uudaryy........ hereafter. In the following analyses, we only consider when the mean SFA strength smaller than the boundary, i.e., > O. Since when < O, the bump movement is in the travelling wave state for most of the time and no longer stochastie process. The second part, / (2//Tam) Em.o(!), reflects the fluctuations of the SFA strength over time, and we denote y = om/ (2 Tam) as the noise-to-strenge ratio hereafter. Note that if the SFA strength fixed, i.c.. 7 = Eq. (15) degenerates into an Ornstein-Uhlenbee (OU) process (37]. In such a case, the stationary distribution of s(t) has a Gaussian form, which leads the Brownian motion of the bump position ztt).
Non-Brownian motion occurs when the SFA strength fluctuates, i.e., the noise-to-stee tosteett- ratio y 0. In such case, can obtain the stationary distribution of s(t) by solving the corresponding Fokker-Plande equation (see SI.2 for the details), which gives,
p( co (o² + +728?) (1+1/77)) for i {1,2]
(16)
where 81 and 82 represent the coordinates of on the axes of xı and 22, respectively. o²
(Veroou/ (VFrT displacement of z(t) in a short time interval öt is calculated to be ||Az|| msst /
(v2ar)² and cu is normalization constant. From Eq. 14). the
/28t/(TT)oz (owttoooAAAAe /AuEu, Replacing with its stationary distribution given by Eq. (16 derive the distribution of | which is written as,
finally
A(|)) ~ | -1-
(17)
which satisfies the power-law distribution (see SI.3 for the details). Comparing Eq. with Eq. 1I
we obtain the Lévy exponent of the bump movement,
2
a = 1 1+
(18)

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It shows that the Lévy exponent is determined by two factors, which are the distance-to toaunne--t-haeee -ootancc-to-bouddayy and the noise-to-strength ratio 7..
Based on the above theoretical analyses and comprehensive simulations, we investigate the effects of y and Y on the mobility of the bump state in the attractor space, and the results are summarized in Fig. 3 To elucidate the mechanism of generating Lévy flights clearly, we inspect the effects of j and y separately by varying only one of them each time while fixing the other. The results are introduced
below.
Noise-to-se tee--ttssttegthh ratlo 3
nights
Levy
200
Simullation time steps
200
Simulation time steps
1.S Lovy exponent a
Travelling wave state
130
100
Brownian
.02 0.1 D 0.1 02
motion
7-000,4-0.0
1:066
Distance to bouundayry
(D)
(F)
2.0
2.0
Levy exponent
1.8
Optimal seaacch
Levy expooent
1.6
1.6
1.4
1.4
simuaation resulls
simulation
theoretical results
theoretical results
1.0
1.0
-1.0
Distanceto-bouuddary P
Distane-tt-boonddayy
Nolse-to-strength ration
Lévy flights in a two-dimensional CANN. (A) The phase diagram of the network with respect to the distance-to-boundaar toreee-rr-oouurrrroooeoo / and the noise-to-strength ratio 7. (B) An example of Brownian motion in the attractor space with μ = 0.95 and 7 = 0.6, corresponding to the blue point in (A). The inset is close-up of the local Brownian motion. (C) An example of Lévy flights in the attractor space with μ = 0.05 and y = 0.6, corresponding to the orange point in (A). (D) The Lévy exponent a vs. with 7 = 0.9, corresponding the horizontal dashed line in (A). Note that when μz 72 /2/ i.e., 4 > 0.405, all a > 2 (Brownian motion) will converge to a = 2 due to the Central Limit Theorem. (E) The search efficiency 1 vs. μ.. The optimal search is achieved when μ - 0, indicated by the orange point. (F) The Lévy exponent a vs. y with μ = 0.1, corresponding to the vertical dashed line in (A). For the setting of other parameters, see $1.4 for the details.
Figure 3:
The effect of the distunce-t- p. Fixing the boise-to-strength ratio y while varying μ. we have the following observations: a) In the case of μ 2 72/2. which gives 2 2, the bump movement displays Brownian motion (Fig. 33). This because when μu 2 /2/2, 2, the SEA strength is far away from the boundary, which is too weak to generate the travelling wave state. Thus, the bump movement is mainly driven by noise fluctuations in the neural dynamics (Eq. (2)). b) In the case of 0 < p < 72/2, which gives a < 2, the bump movement displays Lévy flights (Fig. 3C). In this parameter regime, decreasing will decrease the Lévy exponent a from 2 to gradually 3D). This is because m gets closer to the travelling wave boundary, adaptation noises become more likely to push the SFA strength to cross the boundary, and hence long-jump movement occurs more frequent compared to local Brownian motion. When m eventually approaches the travelling wave boundary, i.c., + O, which gives a + 1, the network achieves the optimal Lévy search in the attractor space. Note that in the simulation, power-law distribution, as very large jumps exceeding the size of the attractor space are excluded. This explains why the simulated Lévy exponent slightly larger than the theoretical value of (see Fig. (D). c) In the case of μ < O. the network dynamics will be dominated by the travelling wave state, which overrides the effects both neuronal and adaptation noises, and the bump movement no longer stochastic. d) We observe that the search efficiency of the bump (measured by the number of locations in the attractor space visited by the bump a unit distance) reaches the optimal value when μ 0 (Fig. B),, manifesting the characteristics of Lévy flights.
can only estimate the value of from truncated
7

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The effect of the noise-to-signal ratio Fixing the distance (c.g, P. 0.1, which is close the travelling wave boundary) while varying 7, we have the following observations: a) In the case of y I 22μ, which gives 2 2, the bump movement displays Brownian motion. This understandable: although the mean SFA strength is close to the travelling wave boundary, the adaptation noises are too small to push the SFA strength to cross the boundary. and hence the bump movement is mainly driven by noise fluctuations the neural dynamics (Eq, (2)). ) b) In the case of /2p, which gives that adaptation noises have higher chances to push the SFA strength to cross the boundary, and hence the Lévy exponent will keep decreasing, until it approaches the value of (Fig. B F.
< 2, the bump movement displays Lévy flights. Increasing y implies
3 Lévy flights in free memory retrieval
(A)
(C)
Festure space
Feature space
Frequency
Hebbian learning
Retrieval trajectory
Feature space x
Feature space
0.01
Searching step size per unit time
Index of retreved items (D)
simulation results
Rebreval Intervais
curve fitting, =0.73
100
80
Retrieval
80
5000 10000
15000
Simulation time steps
Index of retrieved ilems
log(Retrieval intervals)
Figure 4: Modelling Lévy flights in free memory retrieval with an attractor network modulated by noisy adaptation. (A) Memory items are randomly distributed on the two-dimensional feature space and neuronal connections are determined by Hebbian learning. (B) An example of the retrieval trajectory in the feature space. (C) The histogram of step sizes of the retrieval trajectory in (B), which follows a power-law tailed distribution. (D) The index of retrieved items vs. the simulation time. Retrieved items burst intermittently over time. (E) Retrieval intervals vs. the index of retrieved items. Long intervals grow exponentially with interruptions by bursts of short retrieval intervals. (F) The log-log plot of the frequency distribution of retrieval intervals, with estimated Lévy exponent a = 0.73. For the setting of other parameters, see SI.4 for the details.
The first study of Lévy flights in human cognition was carried out by Rhodes and Turvey with free memory retrieval task [1I]. In the experiment, eight participants were asked to verbally recall animal names as many as possible (without repetition) within about minutes, and the time intervals between successive recalls were recorded. Intriguingly, they found an exponential increase of long retrieval intervals that are interrupted by bursts of short intervals. Overall, the retrieval intervals can be described by Lévy flights with exponents of 0.37 S a S 0.98 for different participants. Here, we show that this Lévy-fight behavior free memory retrieval can be described by random search on a semantic graph which is modelled by attractor network with noisy adaptation.
Specifically, build a Hopfield-like attractor network, in which each memory item is represented by a localized neural population randomly distributed in -wo-dimensional feature space. The neuronal connections are determined by Hebbian learning, such that each memory item is encoded as an attractor of the network (Fig. [A). Successful retrieval of memory item means that the network state falls into the corresponding attractor. Note that such network, the stationary states no longer form a continuous manifold of neutral stability as in CANN. Even so, adaptation implemented by SFA can still drive the network state to travel among different attractors if the adaptation strength strong enough. Thus, when the mean and variance of the SFA strength are set properly, the network state

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switches intermittently between local motion and long-jump motion in the feature space, exhibiting the characteristics of Lévy flights. This is confirmed by the simulation results in Fig. AB&C.
We have shown that when the memory recall in the human brain is treated as a random search on a semantic graph on which memory items uniformly distributed, the whole process can be described as Lévy flights in the attractor space of a Hopfield-like attractor network. To reproduce the experiment findings which show that the time intervals between successive recalls follow long-tail power-law distribution, we further convert the Lévy flights in the 2-dimension space as shown in Fig. A into their counterparts in the temporal domain. For this purpose, we calculate the time intervals between successive retrievals which are measured by the duration of the network state travelling from one attractor to another (note that each attractor can only be visited once as required in the human experiment, and the recall is terminated once the network state visits the same attractor again due to local motion). Fig. 4 shows that the number of retrieved items scales logarithmically with the simulation time, and memory items are recalled in burst intermittentlyy, which agrees well with the experimental finding (see Fig. in II]. Further analysis shows that long retrieval intervals are interrupted by bursts of very short intervals and the length of long intervals increases exponentially over time (Fig. 4.). which agrees very well with the experimental finding (see Fig. la in [.]. We also calculate the Lévy exponent by a linear fitting of the log-log frequency distribution of retrieval intervals and obtain a = 0.73 (Fig. 4 which is in the range of the experimental finding 0.37 a S 0.98. It is also worth noting that this shows the temporal property of Lévy flights (retrieval intervals). while the a (not shown) which can be obtained in Fig. 4C shows the spatial property of Lévy flights (step sizes on the graph).
4 Conclusion and Discussion
attractor network model to elucidate the mechanism of generating Lévy flights at the circuit level in neural systems. By analyzing the dynamies of CANN with noisy SFA, we find that when the mean SFA strength is set close to the travelling wave boundary, noises in adaptation can easily cause the network state to intermittently switch between local motion and long-jump motion, displaying Lévy-flight patterns in the attractor space. We theoretically derive the power-law distribution of the step sizes of the bump movement, and show that the Lévy exponent is determined by the joint effect of two key factors, i.e. the distance of the mean SFA strength to the travelling wave boundary, and the noise level in the adaptation. We also demonstrate that optimal search in the attractor space achieved when the mean SFA strength approaches the travelling wave boundary. Simulation results agree with our theoretical analyses very well.
In the present study, we have studied noisy-adaptation-modules
Furthermore, we generalize the theoretical findings to a Hopfield-like attractor network and use them to explain the Lévy-flight patterns found in free memory retrieval. The Hopfield-like attractor network encompasses a low -ow-dimensional attractor space which defines semantic graph of memory items, and movement of the network state in the attractor space corresponds to a random mental exploration on the semantic graph. By introducing noisy adaptation into the neural dynamics, we observe that the network exhibits Lévy-flight moving patterns the attractor space, which reproduce the experimental findings very well, including bursts of retrieved items during the retrieval process, the exponential increase of long retrieval intervals over time, and the value of the Lévy exponent.
Related works of generating Lévy flights in the brain. Mathematical models for demonstrating Lévy flights in movement patterns of living things have been intensively studied in the literature, ranging from sub-cellular [38] 39), to the individual organism 7) and to population scales 440 41). Typically, these studies built up phenomenological models which describe the time evolution of the diffusion and transport of moving object, e.g., by coupled continuous time random walks (CTRW) 42 [8], by various fractional PDEs 43 44 and by integro-differential equations 45 46
Up to now, mechanisms of generating Lévy flights in the brain (especially at the neural circuit level), which underlie the generation of macroscopic Lévy-flight behaviors across different levels of living organisms, have been rarely studied. Very recently, two works have drawn our attention. First is the work from McNamee et al. [47, in which they proposed an antorhinal-hippoccan with linear feedback that implements spectral modulation of sequence generation. Their model can explain the Lévy-fighht patterns found in foraging behaviors and other cognitive functions in animals.
network model
However, it is not a circuit model based on detailed neural dynamics, but instead based on master
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equation that describes the time evolution of the state distribution in the network. In our work, we start from detailed continuous attractor network model with generic neural features. We derive the dynamics of the network state in the attractor space and carry out theoretical analysis to elucidate how neural adaptation causes Lévy flights in the network. These detailed analyses enables us to show the relationships between the critical parameters in our model (i.e., the distance-tor coannce-t--oudddryy y and the noise-to-signe signal ratio 7) and the properties of Lévy flights quantitatively. We expect that this detailed computational model can give us more insight into understanding the generation of Lévy flights in the brain and guide us to carry out further experimental and theoretical studies.
The second related work is from Gong and his co- authhrooo [88 49 50] In the study [48]. Wardak and Gong proposed a leaky integrate and fire (LIF) neuron model with heterogeneous connection strengths. and demonstrated that the membrane potential of the LIF neuron undergoes Lévy flights. They carried out mathematical analysis to show that the total synaptic input to the LIF neuron has nonequilibrium Lévy noises when the distribution of the heterogeneous synaptic connections has power-law tail, and thus the membrane potential probability density satisfies a fractional Fokker- Kaakker-llack equation. Liu et al. further extend the model to balanced spiking neural circuit model to explain the Lévy-flight patterns observed in the gamma bursts in the area MT of marmoset monkeys [50]. Notably, they also found that the Lévy-flight patterns observed in their model arise from the critical transitions between the asynchronous and propagating wave states, similar to the transitions between the localized bump and travelling wave states in our model. Although both their and our studies observed the Lévy-flight -fightt phenomenon in neural networks, the models and the associated mechanisms for generating Lévy flights are very different. Specifically, their model considers power-law tailed synaptic connections, which lead to Lévy noises in a LIF neuron. Theoretical analyses showed that the membrane potential can then be described by fractional Fokker- equation with Lévy noises, which is well- known mathematical strategy to generate Lévy flights [51 52 53) On the other hand, we consider an attractor neural network with noisy adaptation, and the network state movement is described by a Fokker-Plancl equation with multiplicative Gaussian noises, which belongs to another mathematical strategy to generate Lévy flights [54] [53]. It will be interesting explore in future works whether one mechanism is more biologically plausible or whether both mechanisms exist in the brain but are applied in different situations.
Other potential mechanisms of generating Lévy flights in attractor models. The mechanism we propose exploits noisy adaptation to generate Lévy flights. Both noises and adaptation are the intrinsic properties ubiquitously observed in neural systems, and hence they provide an internal strategy to generate Lévy flights without relying on external inputs. This based mechanism is appealing for efficient information searching in an unknown environment with scarce targets, e.g., for animal foraging an open space with sparse prey. Notably, recent research has revealed that when targets are abundantly distributed in the environment, animals can switch the search strategy from Lévy flights to Brownian- motion [4 (5). Our model can also account for this strategy switch phenomenon. As shown in Fig. AA by varying the SFA strength, which could be realized by neural modulation in reality, the network dynamics can switch between Brownian motion and Lévy flights. In addition to SFA, neural systems have other mechanisms to implement adaptive responses, including, for instance, the short-term depression (STD) of synapses between neurons (57) and the long-range feedback inhibition between cortical areas 58). These mechanisms can also destabilize the attractor states of neural circuit via negative feedback, and generate Lévy flights as analyzed in this study. The differences in these mechanisms are that they operate at different spatial and temporal scales. For instance, SFA occurs at the single neuron level and has a time scale of less than one hundred milliseconds, STD occurs at the synapse level and has time scale of hundreds to thousands milliseconds, and the long-range feedback inhibition occurs between brain regions and has time scale of tens of milliseconds. Thus, the brain may employ different strategies generate Lévy-flight patterns at different temporal scales and in different brain regions, and they support efficient information processing in different cognitive functions. It will be interesting to explore these issues in our future work.
Acknowledgement
This work was supported by Beijing Academy of Artificial Intelligence (BAAI), Guangdong Province with grant (No. 201 8B030338001, S. Wu), Huawei Technology Co.,Ltd (No. YBN2019105137 S. Wu) and the National Natural Science Foundation of China (No. 4861 425025, T.J.Hungg.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations your work? [Yes] Please see Sec. 4 for the detailed discussions.
(c) Did you discuss any potential negative socictal impacts of your work? [No] Our study is fundamental on brain science and not tied to applications.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Please see Sec. 2.2
(b) Did you include complete proofs of all theoretical results? [Yes] Please Sec. and many detailed proofs are presented in the Supplementary Information.
2.2
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed reproduce the main ex- perimental results (either in the supplemental material as a URL)? [Yes] We have included the code for reproducing the main results in the supplemental material.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] For the setting of hyperparameters, please see SI.4 in the Supplementary Information.
(c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)? [Yes] We only need to include error bars in two sub-figures in Fig. 3 and we have already done it.
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] We do not GPU CPU clusters. It is sufficient to run our code on a laptop-
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14


Reviews:
[{"review_id": "YR5MCZDig4P", "reviewer": "Reviewer_FVLu", "summary": "The study describes a model of wave propagation in recurrent neural network displaced on a 2D plan. The neural activation is modeled with a simple divise inhibition model, and the lateral connectivity is given by a Gaussian kernel.  The paper reviews some works that have studied the stability of the activation pattern in this setting and the motion if the neuron model has spike frequency adaptation (SFA). It seems that this novelty here is to include noisy in the dynamics of adaptation. By solving this analytically, the authors shows Levy flight dynamics.\n\nBy studying a special case where the recurrent connectivity is derived from Hebbian learning, it is argued that it could model jumpy wandering thoughts.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Overall the paper is very well written. The review of previous works is very clear even for me although I did not know about CANNs.\n\nMany of the intermediate steps of the mathematical demonstration are inherited from other papers and the rest is detailed in the appendix. I tried to review some of that and I did not find problems but I did not go through the entire supplementary materials.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "One limitation is that the Levy-Flights seem to appear only in the model when the adaptation is noisy. This seems to be a strong requirement because the chosen model of adaptation is simplified and I am convinced whether it captures well the wide range of adaptation mechanisms and their variety. I am not sure for instance that this model of adaptation is as solidly grounded on data as divisive inhibition. So would Levy flights emerge if the adaptation model is slightly different? or noisy in a different way? Perhaps at least in simulation it is possible to explore whether it is sufficient that noisy connectivity pattern or neuron position is enough to produce the effect rather that this particular model of noisy adaptation? I also wonder whether other types of frequency adaptation (for instance non-linear \"spike\" frequency adaptation involving r rather than U) would still lead to this behavior.\n\nA second limitation is that the authors make an implicit far-fetched connection between Levy flights in behavior from Drosophilia and humans and Levy-flights in neural activity (line 35 to 44). The geographical motion of human is certainly explained by a different phenomena than the behavior Drosophilia, is it therefore ok to suggest that your model is an answer to all this by mentioning it in the intro? I believe it would be more convincing  to minimize or clarify this paragraph, at least I do not think that the presented model explains all these behavior. I do find that the speculation that the model could explain mind wandering is charming, although quite hypothetical. Ideally, this third section should acknowledge that this connection is a speculation or justify it better.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4 hours"}, "extra_text": {}}, {"review_id": "T1DHqRNCZ9L", "reviewer": "Reviewer_TH9t", "summary": "The submission suggests a neural mechanism underlying Levy flights in continuous attractor neural networks (CANN). More specifically, the author(s) proposes that superdiffusion (as observed in free memory retrieval and foraging behavior) occurs in CANN with noisy neural adaptation. External input noise together with adaptation makes the CANN switch between two regimes, local random walks and long-jump motion, separated in the CANN by the traveling wave boundary. Together, these regimes generate superdiffusion. An analytical treatment in reduced dimensionality by a Fokker-Planck approach reveals the two control parameters controlling the (super)diffusion in the model. Implications for free memory retrieval are discussed in a toy model, and as expected, superdiffusion consistent with experimental evidence is observed.\n\nIn conclusion, the study proposes an exciting new link to generate in a widely used neural network architecture (CANN) experimentally observed Levy flights using a neuronally plausible mechanism and studies it analytically.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "#postrebuttal:\nAll major points were addressed.\n\nOriginality:\nThe submission proposes a simple CANN model that is being reduced to two dynamic variables. I am not an expert in the field, but I have not seen that model before; it combines two common 'ingredients' (adaptation and continuous attractor models) in a mathematically tractable model.\n(Here seems a related work, but I don't have access to it: https://ses.library.usyd.edu.au/handle/2123/18808\nThe abstract says: \"we develop a new type of two-dimensional neural field model that incorporates refractoriness as nonlinear negative feedback. We construct explicit bump solutions and perform a linear stability analysis, which reveals the emergence of stable bump activity as well as a critical transition from the bump state into propagating waves. Using numerical simulation, we show that the neural field exhibits local propagating patterns with rich dynamics including periodic, rotating and chaotic dynamics. We then show that propagating local patterns undergoing Levy flight emerge from a realistic cortical circuit model and that they can account for a wide range of neural response properties during both spontaneous and stimulus-driven activities.\"\n\nEven if this work would be similar, the abstract doesn't say anything about an analytical treatment, so I would consider the presented manuscript still to be sufficiently original. Still, the work should be cited, and similarities and differences should be explained.\nWhile previous work on Levy flights and Levy walks (and anomalous diffusion in general) is cited in the discussion, it would help a lot if the mathematical differences would be explained (e.i. why is it not necessary to use a fractional Fokker-Planck equation and fractional calculus here). \n\nQuality:\nThe model has the appeal of simplicity, so it can be (in a reduced version) be analytically treated using a Fokker-Planck approach, yielding the stationary distributions of s (the separation between bump U and adaptation current V) assuming separation of time scales. Based on that, the phase diagram of 3 dynamics regimes (traveling waves, Brownian motion, and Levy flights) is constructed, and the Levy exponent of the bump movement is determined, which depends on two parameters, 'distance to boundary \\mu and noise-to-strength ratio \\gamma. Their effect is then compared both analytically and numerically, and numerics and analytical calculations approximately agree.\n\nMajor points:\n* What is not so clear to me is to what extent the described model actually exhibits a Levy flight. In my (admittedly non-expert) understanding, in Levy flights, increments are drawn from a Levy distribution which asymptotically follows a power law. In the presented manuscript, I don't see any direct evidence that the increments follow a Levy distribution and/or power law. Figure 1B shows an example, but the y-axis is not log-scales, so it is now clear if the decay follows a power law. Moreover, based on the analytical derivation, it is also not fully transparent where such a power law would come from. In the supplement, where the power-law is derived, besides the assumption of a separation of timescales, the assumption is made that \"gamma an infinitesimal\" (line 121), which, I guess, should read \"gamma is infinitesimal\". But in the rest of the work, gamma is not infinitesimal -this might be a misunderstanding on my side, clarification would be desirable. I also ran the Matlab code in the supplementary material, but it didn't show how the Levy exponent \\alpha reported in the figures was estimated. Also for this, clarification would be desirable. \n* Potentially related to the first point, the main model ( \n* There is no code provided to reproduce the results on free memory retrieval (Figure 4), which seem to be purely numerically in nature.\n* An analysis of how the core characteristics of the network (e.g., the Levy exponent \\alpha) depend on the model parameters is missing. Do you expect the same behavior as in the model corresponding to figure 3? Based on that, can you predict how the empirically observed \\alpha =0.73 would change when you change your model parameters, e.g., input noise \\sigma_U, time constants \\tau and \\tau_V, inhibitory strength k, and Recurrent connection strength J_0? Is it important to have a variance of the SFA strength (\\sigma_m) to make the model work? If so, why? \n* In the analysis of the Hopfield-like network, it is not clear how \\alpha is measured. The manuscript text says: \"Further analysis shows that long retrieval intervals are interrupted by bursts of very short intervals and the length of long intervals increases exponentially over time (Fig. 4E). If the length of long intervals increases over time, does it mean the distribution of intervals changes over time, thus also \\alpha changes over time? Does it mean there is an aging effect? How is this consistent with the Fokker-Planck analysis, where a stationary distribution of s and z is assumed? \n\n\nMinor points:\nOne minor comment on the comparison of analytical and numerical results in Figures 3D and 3F: Please clarify, what the error bars indicate. Are these standard errors of the mean or standard deviation or inter-quartile rate? Also, what ensemble are these error bars calculated over?\n* Figure 3D and 3F report numerical results in orange, it is enough to should the orange data points, the lines between the orange points might create the illusion that there is data, but as I understand, there is none. This might also help to make the blue (analytical results) visible.\nI'd be delighted to be convinced that the limitations mentioned above are just resulting from my lack of understanding and are not intrinsic to the work. \n\n* It would also be interesting to look at the search efficiency as a function of noise-to-strength \\gamma.\n* It would demonstrate that as claimed with a larger grid (when the distribution is less truncated), the numerical simulations approach the theoretical predictions.\n* Could the spontaneous activity in the Hopfield-like attractor network also be generated from spontaneous recurrent activity instead of external input drive (e.g., chaos as in asymmetric Hopfield-type networks)?\n* Figure 3A and Supplementary figure 2C seem to be smoothed, it would be desirable just to display the raw numerical results without any smoothing that might create the illusion of intermediate values where there are none.\n\nSome typos:\n* The label in 3E reads \"effeciency\" but should read \"efficiency\".\n* \"Noises in neural adaptation causes\" should read \"Noise in neural adaptation causes\"\n* \"also termed as anomalous\" should read \"also termed anomalous\"\n* \"agree with our theoretical analyses very well\" should read \"agree with our theoretical analysis very well\"\n* \"The neuronal recurrent connections\" should read \"The recurrent neuronal connections\"\n* \"represent the coordinates of s on the the axes\" should read \"represent the coordinates of s on the axes\"", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "There is no societal impact to discuss.\nLimitations:\nThe authors discuss an extension to STD (short-term depression), but a discussion of the limitations is largely missing.\nHere are some potential limitations:\n* The derivation in the supplementary material (4) seems to be rather heuristic:\n- e.g., the variance of Levy flights is generally diverging; this seems not to be considered here (also not in the supplement).\n- the origin of the power-law (scale-invariance) is unclear; I would assume that the traveling waves have a typical scale. \n- the switching statistics between traveling wave and Brownian motion are not characterized; it is unclear why the length by which the bump travels should be long-tailed.\n- a formal Kramers-Moyal expansion to derive the Fokker-Planck equation is missing.\n* If the authors claim their proposed mechanism to generate Levy flights is beyond the mathematical physics literature of anomalous diffusion, it should be made clear what is different from known mechanisms to generate Levy flights (e.g., fractional Fokker-Planck equation). In case the mechanism proposed here can actually be mapped back to a previously described mechanism (which would be totally fine), this should be clearly stated.\n* An analysis of the dependence of the Levy-flight phenomenon on the parameters of the Hopfield type model (figure 4) is missing.\n* A clear explanation of the link between the model described in figure 3 and the Hopfield-type model (figure 4) is missing. Specifically, while in the CANN, a stationary state was assumed, the Hopfield-type model (figure 4) seems not to be characterized in a stationary state (see above).\nI'd be delighted to be convinced that the limitations mentioned above are just resulting from my lack of understanding and are not intrinsic to the work. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "12"}, "extra_text": {}}, {"review_id": "OY5uD2hXQXJ", "reviewer": "Reviewer_JwRM", "summary": "The paper discusses the appearance of Levy flights in animal behaviour, and attempts to justify the presence of a \"neural mechanism\" for this behaviour. The authors begin with a continuous attractor neural network (CANN) model: a linear stochastic partial differential equation involving an \"adaptive current\" term V, here called \"spike frequency adaptation\" (SFA). This SFA is the noise adaptation component mentioned in the title, and naturally involves another white noise term multiplied by the activity level U. The authors then approximate the dynamics by condensing the SPDE into a system of SDEs through projection onto two modes. One of the resulting SDEs also contains a multiplicative noise term. This results in a stationary distribution with an asymptotic power law. The appearance of this power law in the motion of these modes is claimed to justify the presence of Levy flights in \"neural circuits\", and hence in the brain, supported experimentally by a Hebbian-learning attractor network. ", "questions": "", "limitations": "", "rating": 6, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I do not believe this is the correct venue for this paper. The problems and motivations appear highly scientific, rooted in applied mathematics and neuroscience, and not related to computer science (or, indeed, to neural networks, despite the name of the model used).\n\nAs a mathematician, I recognize that I am not the target audience for this work. From my point of view, when stripped back to its most basic elements, the authors appear to have rediscovered the relationship between multiplicative noise and heavy tails in a particular SPDE model --- see, for example, [1]. Some discussion of this general phenomenon is probably warranted to put the results in context. There is an active area of research surrounding the consequences of this property in ML, including [2,3]. However, I do not see how this paper would be of interest to the ML/statistics community, and think it is better submitted to a neuroscience outlet. \n\nNevertheless, the paper appears grammatically sound. A few minor issues: \n- The CANN SPDE is not ill-posed per se, but Ito/Stratonovich properties of the white noise are not specified\n- The intuition and discussion surrounding Levy flights as combining Brownian motion with long jumps is incorrect. Levy flights generally do not exhibit properties analogous to Brownian motion (unless alpha=2). An alpha-stable Levy process is one example of a Levy flight and is purely discontinuous with no Brownian motion part. - \"Fokker-Plank\" should be written \"Fokker-Planck\"\n- Figure 1 doesn't seem like an especially fair comparison. The variance of the Brownian motion should be adjusted to give a more appropriate match between the Levy flight and the Brownian motion. \n- In Figure 4, estimating power laws using curve-fitting to a log-log plot should be avoided, as this approach incurs tremendous bias. An MLE-based approach should be considered instead.\n\n\n[1] Biró, T. S., & Jakovác, A. (2005). Power-law tails from multiplicative noise. Physical review letters, 94(13), 132302.\n\n[2] Gurbuzbalaban, M., Simsekli, U., & Zhu, L. (2021). The heavy-tail phenomenon in SGD. In International Conference on Machine Learning (pp. 3964-3975). PMLR.\n\n[3] Hodgkinson, L., & Mahoney, M. (2021, July). Multiplicative noise and heavy tails in stochastic optimization. In International Conference on Machine Learning (pp. 4262-4274). PMLR.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The conclusions are not justified up to the standards of mathematical or statistical rigor expected at this venue. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}, {"review_id": "1kocximV2mR", "reviewer": "Reviewer_1hsR", "summary": "Levy flights are a class of random walks that are efficient search strategies, and human cognitive functions have been shown to exhibit Levy flight characteristics. This work demonstrates that an adaptation of continuous attractor neural networks (CANNs) can generate Levy flight dynamics. Essentially, by setting the parameters of the network near a threshold where the network transitions from Brownian dynamics to traveling wave dynamics, and adding external noise, one can induce the dynamics to randomly switch between Brownian dynamics and traveling wave dynamics, thus mimicking Levy flight dynamics.", "questions": "", "limitations": "", "rating": 7, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This work is the first to study how the brain can generate Levy flight patterns which have been observed experimentally. Some of the interesting theory unpinning this work was previously established in [30,31], where it was shown that depending on the parameters, the CANN will exhibit Brownian dynamics or traveling wave dynamics. The main novelty here is to show that when the parameters are near the threshold between the 2 modes and there is external noise, the network will switch between the 2 modes and they obtain an analytical expression for the Levy exponent. In the conclusion, they propose that short-term depression can destabilize attractor states \n\nMinor comments:\n\n1. Following Eq. (2), should \"Fig. 2B\" be \"Fig. 2A\"?\n2. Caption of Fig. 2, I believe the authors mean \"for clarity\" instead of \"for clearance\".", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "How would one test whether brains use \"spike frequency adaption\" to generate Levy flights?", "review_text": "", "extra_scores": {"ethical_concerns": "I have no ethical concerns.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}]

Author rebuttal:

Meta-review: All the reviewers have agreed that the submission is original and interesting with clear contributions to the domain. Authors’ responses also addressed most major concerns. Hence, I recommend an acceptance.

On another note, as reviewer JwRM partially pointed out, using power-laws and Levy flights for modeling stochastic optimization algorithms have attracted considerable attention in other parts of ML as well. In this respect, I suggest the authors to check the papers and discuss in their paper if they think relevant, that reviewer JwRM suggested, as well as the following one, which is directly based on a Levy-driven Langevin equation:

Simsekli, U., Sagun, L., & Gurbuzbalaban, M. (2019, May). A tail-index analysis of stochastic gradient noise in deep neural networks. In International Conference on Machine Learning (pp. 5827-5837). PMLR.

Zhou, P., Feng, J., Ma, C., Xiong, C., & Hoi, S. C. H. (2020). Towards Theoretically Understanding Why Sgd Generalizes Better Than Adam in Deep Learning. Advances in Neural Information Processing Systems, 33.

Decision: Accept (Poster)