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# Drift–diffusion models for multiple-alternative forced-choice decision making

- Alex Roxin
^{1, 2}Email author

**9**:5

https://doi.org/10.1186/s13408-019-0073-4

© The Author(s) 2019

**Received:**9 January 2019**Accepted:**10 June 2019**Published:**3 July 2019

## Abstract

The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift–diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift–diffusion process to multiple alternatives. The competition between *n* alternatives takes place in a linear subspace of \(n-1\) dimensions; that is, there are \(n-1\) decision variables, which are coupled through correlated noise sources. I derive the multiple-alternative DDM starting from a system of coupled, linear firing rate equations. I also show that a Bayesian sequential probability ratio test for multiple alternatives is, in fact, equivalent to these same linear DDMs, but with time-varying thresholds. If the original neuronal system is nonlinear, one can once again derive a model describing a lower-dimensional diffusion process. The dynamics of the nonlinear DDM can be recast as the motion of a particle on a potential, the general form of which is given analytically for an arbitrary number of alternatives.

## Keywords

- Decision making
- Networks
- Winner-take-all

## 1 Introduction

Perceptual decision-making tasks require a subject to make a categorical decision based on noisy or ambiguous sensory evidence. A computationally advantageous strategy in doing so is to integrate the sensory evidence in time, thereby improving the signal-to-noise ratio. Indeed, when faced with two possible alternatives, accumulating the difference in evidence for the two alternatives until a fixed threshold is reached is an optimal strategy, in that it minimizes the mean reaction time for a desired level of performance. This is the computation carried out by the sequential probability ratio test devised by Wald [1], and its continuous-time variant, the drift–diffusion model (DDM) [2]. It would be hard to overstate the success of these models in fitting psychophysical data from both animals and human subjects in a wide array of tasks, e.g. [2–5], suggesting that brain circuits can implement a computation analogous to the DDM.

At the same time, neuroscientists have characterized the neuronal activity in cortical areas of monkey, which appear to reflect an integration process during DM tasks [6], although see [7]. The relevant computational building blocks, as revealed from decades of in-vivo electrophysiology, seem to be neurons, the activity of which selectively increases with increasing likelihood for a given upcoming choice. Attractor network models, built on this principle of competing, selective neuronal populations, generate realistic performance and reaction times; they also provide a neuronal description which captures some salient qualitative features of the in-vivo data [8, 9].

While focus in the neuroscience community has been almost exclusively on two-alternative DM (although see [10]), from a computational perspective there does not seem to be any qualitative difference between two or more alternatives. In fact, in a model, increasing the number of alternatives is as trivial as adding another neuronal population to the competition. On the other hand, how to add an alternative to the DDM framework does not seem, on the face of things, obvious. Several groups have sought to link the DDM and attractor networks for two-alternative DM. When the attractor network is assumed linear, one can easily derive an equation for a decision variable, representing the difference in the activities of the two competing populations, which precisely obeys a DDM [11]. For three-alternative DM previous work has shown that a 2D diffusion process can be defined by taking appropriate linear combinations of the three input streams [12]. The general *n*-alternative case for leaky accumulators has also been treated [13]. In the first section of the paper I will summarize and build upon this previous work to illustrate how one can obtain an equivalent DDM, starting from a set of linear firing rate equations which compete through global inhibitory feedback. The relevant decision variables are combinations of the activity of the neuronal populations, and which represent distinct modes of competition. Specifically, I will propose a set of “competition” basis functions which allow for a simple, systematic derivation of the DDMs for any *n*. I will also show how a Bayesian implementation of the the multiple sequential probability ratio test (MSPRT) [14–16] is equivalent in the continuum limit to these same DDMs, but with a moving threshold.

Of course, linear models do not accurately describe the neuronal data from experiments on DM. However, previous work has shown that attractor network models for two-alternative DM operate in the vicinity of pitchfork bifurcation, which is what underlies the winner-take-all competition leading to the decision dynamics [17]. In this regime the neuronal dynamics is well described by a stochastic normal-form equation which right at the bifurcation is precisely equivalent to the DDM with an additional cubic nonlinearity. This *nonlinear* DDM fits behavioral data extremely well, including both correct and error reaction times. In the second part of the paper I will show how such normal-form equations can be derived for an arbitrary number of neuronal populations. These equations can be thought of as nonlinear DDMs and, in fact, are identical to the linear DDMs with the addition of quadratic nonlinearities (for \(n>2\)). Amazingly, the dynamics of such a nonlinear DDM can be recast as the diffusion of particle on a potential, which is obtained analytically, for arbitrary *n*.

## 2 Results

*X*is the decision variable, and

*μ*is the drift or the degree of evidence in favor of one choice over the other: we can associate choice 1 with positive values of

*X*and choice 2 with negative values. The Gaussian process \(\xi (t)\) represents noise and/or uncertainty in the integration process, with \(\langle \xi (t)\rangle = 0\) and \(\langle \xi (t)\xi (t^{\prime })\rangle =\sigma ^{2}\delta (t-t ^{\prime })\). I have also explicitly included a characteristic time scale

*τ*, which will appear naturally if one derives Eq. (1) from a neuronal model. The decision variable evolves until reaching one of two boundaries ±

*θ*at which point a decision for the corresponding choice has been made.

It is clear that a single variable can easily be used to keep track of two competing processes by virtue of its having two possible signs. But what if there are three or more alternatives? In this case it is less clear. In fact, if we consider a drift–diffusion process such as the one in Eq. (1) as an approximation to an actual integration process carried out by neuronal populations, then there is a systematic approach to deriving the corresponding DDM. The value of such an approach is that one can directly tie the DDM to the neuronal dynamics, thereby linking behavior to neuronal activity.

I will first consider the derivation of a DDM starting from a system of linear firing rate equations. This analysis is similar to that found in Sect. 4.4 of [13], although the model of departure is different. In this case the derivation involves a rotation of the system so as to decouple the linear subspace for the competition between populations from the subspace which describes non-competitive dynamical modes. This rotation is equivalent to expressing the firing rates in terms of a set of orthogonal basis functions: one set for the competition, and another for the non-competitive modes. I subsequently consider a system of nonlinear firing rate equations. In this case one can once again derive a reduced set of equations to describe the decision-making dynamics. The reduced models have precisely the form of the corresponding DDM for a linear system, but now with additional nonlinear terms. These terms reflect the winner-take-all dynamics which emerge in nonlinear systems with multi-stability. Not only do the equations have a simple, closed-form solution for any number of alternatives, but they can be succinctly expressed in terms of a multivariate potential.

### 2.1 Derivation of a DDM for two-alternative DM

*s*represents the strength of excitatory self-coupling and

*c*is the strength of the global inhibition. The characteristic time constants of excitation and inhibition are

*τ*and \(\tau _{I}\) respectively. A choice is made for 1 (2) whenever \(r_{1} = r_{th} \) (\(r_{2} = r_{th} \)).

*X*, while \(\mathrm{m}_{C}\) and \(\mathrm{m}_{I}\) stand for the common mode and inhibitory mode respectively.

*X*are found by projecting onto \(\mathbf{e}_{1}\), namely and similarly the dynamics for \(\mathrm{m}_{C}\) and \(\mathrm{m}_{I}\) and found by projecting onto \(\mathbf{e}_{C}\) and \(\mathbf{e}_{I}\) respectively. Doing so results in the set of equations

*X*describes a drift–diffusion process. It is formally identical to Eq. (1) with \(\mu = \frac{I_{1}-I_{2}}{2}\) and \(\xi (t) = \frac{\xi _{1}(t)-\xi _{2}(t)}{2}\). Importantly,

*X*is uncoupled from the common and inhibitory modes, which themselves form a coupled subsystem. For \(s\ne 1\) the decision variable still decouples from the other two equations, but the process now has a leak term (or ballistic for \(s>1\)) [18]. It has therefore been argued that obtaining a DDM from linear neuronal models requires fine tuning, a drawback which can be avoided in nonlinear models in which the linear dynamics is approximated via multi-stability; see e.g. [19, 20]. If one ignores the noise terms, the steady state of this linear system is \((X,\mathrm{m}_{C},\mathrm{m}_{I}) = (X_{0},\mathrm{M}_{c},\mathrm{M} _{I})\) where

### 2.2 Derivation of a DDM for three-alternative DM

I will go over the derivation of a drift–diffusion process for three-choice DM in some detail for clarity, although conceptually it is very similar to the two-choice case. Then the derivation can be trivially extended to n-alternative DM for any n.

*n*alternatives I will always take the first \(n-2\) basis vectors to be those from the \(n-1\) case. The last eigenvector must be orthogonal to these. Specifically, for \(n = 3\), \(\mathbf{r} = \mathbf{e}_{1}X _{1}(t)+\mathbf{e}_{2}X_{2}(t)+\mathbf{e}_{C}\mathrm{m}_{C}(t)+ \mathbf{e}_{I}\mathrm{m}_{I}\), where

### 2.3 Derivation of DDMs for n-alternative DM

*k*th decision variable is given by

*n*-alternative DM is to take for the

*k*th eigenvector

*k*appears in the \((k+1)\)st spot, and which is a generalization of the eigenvector basis taken earlier for two- and three-alternative DM. With this choice, the equation for the

*k*th decision variable can be written

*i*th

*neuronal*population can then be expressed in terms of the decision variables as

*i*th neuronal population wins when \(-(i-1)x_{i-1}+\sum_{l = i}^{n-1}x_{l}+\mathrm{M}_{C} > r _{th}\).

### 2.4 DDM for multiple alternatives as the continuous-time limit of the MSPRT

In the previous section I have illustrated how to derive DDMs starting from a set of linear differential equations describing an underlying integration process of several competing data streams. An alternative computational framework would be to apply a statistical test directly to the data, without any pretensions with regards to a neuronal implementation. In fact, the development of an optimal sequential test for multiple alternatives followed closely on the heels of Wald’s work for two alternatives in the 1940s [22]. Subsequent work proposed a Bayesian framework for a multiple sequential probability ratio test (MSPRT) [14–16]. Here I will show how such a Bayesian MSPRT is equivalent to a corresponding DDM in the continuum limit, albeit with moving thresholds.

*n*possible alternatives, and that the instantaneous evidence for an alternative

*i*is given by \(z_{i}(t)\), which is drawn from a Gaussian distribution with mean \(I_{i}\) and variance \(\sigma ^{2}\). The total observed input over all alternatives

*n*and up to a time

*t*is written

*I*and the hypothesis that alternative

*i*has the highest mean \(H_{i}\). Then, using Bayes theorem, the probability that alternative

*i*has the highest mean given the total observed input is

*i*having the largest mean up to a time

*t*is \(L_{i}(t) = \ln {\Pr (H_{i}|I)}\), which is given by

*i*is given by

*i*should be chosen if the log-likelihood exceeds a given threshold, namely if

*i*, namely that the mean of the distribution is greatest for alternative

*l*, then the expected dynamics of the common mode at long times are

On the other hand, an equivalent DDM with a fixed threshold has worse accuracy and shorter reaction times. The way I choose the parameters for a fair comparison is to set the fixed threshold such that the mean reaction time is identical to the Bayesian model for zero coherence. Another important difference is that the error RTs are longer than the correct RTs for the Bayesian model, see Fig. 2B, an effect which is commonly seen in experiment (and is also reproduced by the nonlinear DDMs studied in the next section of the paper) [17]. On the other hand correct and error RTs for the DDMs are always the same.

## 3 Derivation of a reduced model for two-choice DM for a nonlinear system

*ϕ*(\(\phi _{I}\)) does not need to be specified in the derivation. The noise sources \(\xi _{i}\) are taken to be Gaussian white noise and hence must sit outside of the transfer function; they therefore directly model fluctuations in the population firing rate rather than input fluctuations. Input fluctuations can be modeled by allowing for a non-white noise process and including it directly as an additional term in the argument of the transfer function. Note that here I assume the nonlinearity is a smooth function. This is a reasonable assumption for a noisy system such as a neuron or neuronal circuit. Non-smooth systems, such as piecewise linear equations for DM, require a distinct analytical approach; see, e.g., [23].

*λ*. These eigenvalues are

### 3.1 A brief overview of normal-form derivation

At this point it is still unclear how one can leverage this linear analysis to derive a DDM. Specifically, and unlike in the linear case, one cannot simply rotate the system to uncouple the competition dynamics from the non-competitive modes. Also, note that the steady states in a nonlinear system depend on the external inputs, whereas that is not the case in a linear system. In particular, the DDM has a drift term *μ* which ought to be proportional to the difference in inputs \(I_{1}-I_{2}\), whereas to perform the linear stability we assumed \(I_{1} = I_{2}\). Indeed, if one assumes that the inputs are different, then the fixed-point structure is completely different. The solution is to assume that the inputs are only slightly different, and formalize this by introducing the small parameter *ϵ*. Specifically, we write \(I_{1} = I_{0}+\epsilon ^{2}\Delta I +\epsilon ^{3}\bar{I}_{1}\), and \(I_{2} = I_{0}+\epsilon ^{2}\Delta I +\epsilon ^{3}\bar{I}_{2}\). In this expansion, \(I_{0}\) is the value of the external input which places the system right at the bifurcation in the zero-coherence case. In order to describe the dynamics away from the bifurcation we also allow the external inputs to vary. Specifically, Δ*I* represents the component of the change in input which is common to both populations (overall increased or decreased drive compared to the bifurcation point), while \(\bar{I}_{i}\) is a change to the drive to population *i* alone, and hence captures changes in the coherence of the stimulus. The particular scaling of these terms with *ϵ* is enforced by the solvability conditions which appear at each order. That is, the mathematics dictates what these are; if one chose a more general scaling one would find that only these terms would remain.

*ϵ*and written

*X*is the decision variable which evolves on a slow-time scale, \(T = \epsilon ^{2}t\). The slow-time scale arises from the fact that there is an eigenvector with zero eigenvalue; when we change the parameter values slightly, proportional to

*ϵ*, the growth rate of the dynamics along that eigenvector is no longer zero, but still very small, in fact proportional to \(\epsilon ^{2}\) in this case.

The method for deriving the normal-form equation, i.e. the evolution equation for *X*, involves expanding Eq. (28) in *ϵ*. At each order in *ϵ* there is a set of equations to be solved; at some orders, in this case first at order \(\mathcal{O}(\epsilon ^{3})\), the equations cannot be solved and a solvability condition must be satisfied, which leads to the normal-form equation.

### 3.2 The normal-form equation for two choices

*X*,

*μ*and

*γ*are

*X*and switch the labels on the neuronal populations, the dynamics is once again the same. This reflection symmetry ensures that all terms in

*X*will have odd powers in Eq. (35) [17]. It is broken only when the inputs to the two populations are different, i.e. by the first term on the r.h.s. in Eq. (35).

As we shall see, the stochastic normal-form equation, Eq. (35), which from now on I will refer to as a nonlinear DDM, has a very different form from the nonlinear DDMs for \(n>2\). The reason is, again, the reflection symmetry in the competition subspace for \(n=2\), which is not present for \(n>2\). Therefore, for \(n>2\) the leading-order nonlinearity is quadratic, and, in fact, a much simpler function of the original neuronal parameters.

## 4 Three-alternative forced-choice decision making

The derivation of the normal form, and the corresponding DDM for three-choice DM differs from that for two-alternative DM in several technical details; these differences continue to hold for n-alternative DM for all \(n\ge 3\). Therefore I will go through the derivation in some detail here and will then extend it straightforwardly to the other cases.

To derive the normal form I once again assume that the external inputs to the three populations differ by a small amount, namely \((I_{1},I _{2},I_{3}) = (I_{0},I_{0},I_{0})+\epsilon ^{2}(\bar{I}_{1},\bar{I} _{2},\bar{I}_{3})\), and then expand the firing rates as \(\mathbf{r} = \mathbf{R}+\epsilon (\mathbf{e}_{1}X_{1}(T)+\mathbf{e}_{2}X_{2}(T) )+\mathcal{O}(\epsilon ^{2})\), where the slow time is \(T = \epsilon t\). Note that the inputs are only expanded to second order in *ϵ*, as opposed to third order as in the previous section. The reason is that the solvability condition leading to the normal-form equation for the 2-alternative case arises at third order. This is due to the fact that the bifurcation has a reflection symmetry, i.e. it is a pitchfork bifurcation and so only odd terms in the decision variable are allowed. The lowest-order nonlinear term is therefore the cubic one. On the other hand, for more than two alternatives there is no such reflection symmetry in the corresponding bifurcation to winner-take-all behavior. Therefore the lowest-order nonlinear term is quadratic, as in a saddle-node bifurcation.

*ϵ*. In this case a solvability condition first arises at order \(\epsilon ^{2}\), which also accounts for the different scaling of the slow time compared to two-choice DM. Note that there are two solvability conditions, corresponding to eliminating the projection of terms at that order onto both of the left-null eigenvectors of . As before, the left-null eigenvectors are identical to the right-null eigenvectors. Applying the solvability condition yields the normal-form equations

### 4.1 A note on the difference between the nonlinear DDM for 2A and 3A DM

The dynamics of the nonlinear DDM for 2A, Eq. (35), depends strongly on the sign of the cubic coefficient *γ*. Specifically, when \(\gamma < 0\) the bifurcation is supercritical, while for \(\gamma > 0\) it is subcritical, indicating the existence of a region of multi-stability for \(\Delta I < 0\). In fact, in experiment, cells in parietal cortex which exhibit ramping activity during perceptual DM tasks, also readily show delay activity in anticipation of the sacade to their response field [24]. One possible mechanism for this would be precisely this type of multi-stability. When \(\Delta I = 0\), i.e. when the system sits squarely at the bifurcation, Eq. (35) is identical to its linear counterpart with the sole exception of the cubic term. For \(\gamma < 0\) the state \(X = 0\) is stabilized. In fact, the dynamics of the decision variable can be viewed as the motion of a particle in a potential, which for \(\gamma < 0\) increases rapidly as X grows, pushing the particle back. On the other hand, for \(\gamma > 0\) the potential accelerates the motion of *X*, pushing it off to ±∞. This is very similar to the potential for the linear DDM with absorbing boundaries. Therefore, the nonlinear DDM for two-alternatives is qualitatively similar to the linear DDM when it is subcritical, and hence when the original neuronal system is multi-stable.

On the other hand, the nonlinear DDM for three alternatives, Eq. (40), has a much simpler, quadratic nonlinearity. The consequence of this is that there are no stable fixed points and the decision variables always evolve to ±∞. Furthermore, to leading order there is no dependence on the mean input, indicating that the dynamics is dominated by the behavior right at the bifurcation.^{1} The upshot is that Eq. (40) is as similar to the corresponding linear DDM with absorbing boundaries as possible for a nonlinear system without fine tuning. This remains true for all \(n>2\).

This also means that neuronal systems with inhibition-mediated winner-take-all dynamics are generically multi-stable for \(n>2\), although for \(n = 2\) they need not be. This is due to the reflection symmetry present only for \(n = 2\).

## 5
*n*-alternative forced-choice decision making

*n*-choice case. Again I start with a set of firing rate equations

*n*alternatives, we take \(n-1\) eigenvectors of the form \(e_{k} = (1,1,\dots ,1,-k,0, \dots ,0)\), for the

*k*th eigenvector (again, the −

*k*sits in the (k+1)-st spot). Therefore one can write

*k*th decision variable for

*n*alternatives:

*n*-alternative DM can all be derived from a single, multivariate function:

*k*th decision variable is simply

*f*is given by

*f*. Noise sources lead to a diffusion along this landscape.

## 6 Discussion

In this paper I have illustrated how to derive drift–diffusion models starting from models of neuronal competition for n-alternative decision-making tasks. In the case of linear systems, the derivation consists of nothing more than a rotation of the dynamics onto a subspace of competition modes. This idea is not new, e.g. [13], although I have made the derivation explicit here, and have chosen as a model of departure one in which inhibition is explicitly included as a dynamical variable. It turns out that a Bayesian implementation of a multiple sequential probability ratio test is also equivalent to a DDM in the continuum limit, albeit with time-varying thresholds.

For nonlinear systems, the corresponding DDM is a stochastic normal form, which is obtained here using the method of multiple-scales [25]. The nonlinear DDM was obtained earlier for the special case of two-alternative DM [17]. For four-alternative DM the nonlinear DDM was obtained with a different set of competitive basis functions [26] to describe performance and reaction time from experiments with human subjects. This led to a different set of coupled normal-form equations from those given by Eq. (42), although the resulting dynamics is, of course, the same. The advantage of the choice I have made in this paper for the basis functions, is that they are easily generalizable for any *n*, leading to a simple, closed-form expression for the nonlinear DDM for any arbitrary number of alternatives, Eq. (42).

An alternative approach to describing the behavior in DM tasks, is to develop a statistical or probabilistic description of evidence accumulation; see, e.g., [1, 13, 15, 22, 27]. Such an approach also often leads to a drift–diffusion process in some limit, as is the case for the Bayesian MSPRT studied here, and see also [28]. In fact, recent work has shown that an optimal policy for multiple-alternative decision making can be approximately implemented by an accumulation process with time-varying thresholds, similar to the Bayesian model studied in this manuscript [29]. From a neuroscience perspective, however, it is of interest to pin down how the dynamics of neuronal circuits might give rise to animal behavior which is well described by a drift–diffusion process. This necessitates the analysis of neuronal models at some level. What I have shown here is that the dynamics in a network of *n* neuronal populations which compete via a global pool of inhibitory interneurons, can in general be formally reduced to a nonlinear DDM of dimension \(n-1\). The nonlinear DDMs differ from the linear DDMs through the presence of quadratic (or cubic for \(n = 2\)) nonlinearities which accelerate the winner-take-all competition. In practical terms this nonlinear acceleration serves the same role as the hard threshold in the linear DDMs. Therefore the two classes of DDMs have quite similar behavior.

The DDM is one of the most-used models for fitting data from two-alternative forced-choice decision-making experiments. In fact it provides fits to decision accuracy and reaction time in a wide array of tasks, e.g. [2, 3, 30]. Here I have illustrated how the DDM can be extended to *n* alternatives straightforwardly. It remains to be seen if such DDMs will fit accuracy and reaction times as well as their two-alternative cousin, although one may refer to promising results from [12] for three alternatives. Note also that the form of the nonlinear DDMs, Eqs. (44) and (45) does not depend on the details of the original neuronal equations; this is what is meant by a normal-form equation. The only assumptions needed for the validity of the normal-form equations are that there be global, nonlinear competition between *n* populations. Of course, if the normal form is derived from a given neuronal model, then the parameters *a* and *b* of the nonlinear potential Eq. (44) will depend on the original neuronal parameters.

As stated earlier, the nonlinear DDMs can have dynamics quite similar to the standard, linear DDM with hard thresholds. Nonetheless, there are important qualitative differences between the two classes of models. First of all, both correct and error reaction-time distributions are identical in the linear DDMs, given unbiased initial conditions, whereas the nonlinear DDMs generically show longer error reaction times [17], also a feature of the Bayesian MSPRT. Because error reaction times in experiment indeed tend to be longer than correct ones, the linear DDM cannot be directly fit to data. Rather, variability in the drift rate across trials can be assumed in order to account for differences in error and correct reaction times; see, e.g., [30]. Secondly, nonlinear DDMs exhibit intrinsic dynamics which reflect the winner-take-all nature of neuronal models with strong recurrent connectivity. As a consequence, as the decision variables increase (or decrease) from their initial state, they undergo an acceleration which does not explicitly depend on the value of the external input. This means that the response of the system to fluctuations in the input is not the same late in a trial as it is early on. Specifically, later fluctuations will have lesser impact. Precisely this effect has been seen in the response of neurons in parietal area LIP in monkeys in two-alternative forced-choice decision-making experiments; see Fig. 10B in [31]. Given that network models of neuronal activity driving decision-making behavior lead to nonlinear DDMs, fitting such models to experimental data in principle allows one to link behavioral measures to the underlying neuronal parameters. In fact, it may be that the linear DDM has been so successful in fitting behavioral data over the years precisely because it is a close approximation to the true nonlinear DDM which arises in neuronal circuits with winner-take-all dynamics.

## Declarations

### Acknowledgements

I acknowledge helpful conversations with Klaus Wimmer and valuable suggestions for improvements from the reviewers.

### Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

### Funding

Grant number MTM2015-71509-C2-1-R from the Spanish Ministry of Economics and Competitiveness. Grant 2014 SGR 1265 4662 for the Emergent Group “Network Dynamics” from the Generalitat de Catalunya. This work was partially supported by the CERCA program of the Generalitat de Catalunya.

### Authors’ contributions

All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Competing interests

The author declares to have no competing interests.

### Consent for publication

Not applicable

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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