Summary:

The authors present a continuous attractor neural model which implements (anticipative) tracking. The authors show that spike frequency adaptation (SFA) can induce traveling waves under certain conditions. Interestingly, they show that the effects induced by SFA are similar to those that can be obtained by introducing asymmetric coupling between neurons as in [14], with the advantage that this method does not depend on hard-wired connections. The bulk of the paper is a theoretical analysis of a simplified model and simulations with the complete model for verification. The model can sustain traveling waves and reproduces both perfect tracking and anticipatory tracking.

Comments to the authors:

The paper is well written and interesting - especially the link between the model of [14] and the demonstration that SFA reproduces a similar effect.

I am not an expert in this field, so I find it hard to judge how novel this work is - for instance, you refer to [12], saying that it was already known that a continuous attractor network with SFA can exhibit traveling waves, but in the abstract you state that you propose the SFA mechanism to implement tracking. I think it would be good to clearly delineate what your new contributions are in the introduction.

Are the time constants used in the model (e.g. at figure 2B) realistic? Could you comment on this? A factor 60 difference between \tau and \tau_v seems large to me.

How well does the model reproduce experimental results? Such comparison would make the paper more interesting.


Minor issues:


l. 53. unpredictable flash: I don't think this has to do with the predictability of the flash - it can flash at regular intervals. The point is that the flashing stimulus does not move.

l. 53-55. Despite ...; rephrase -> e.g., Although it is clear that the brain ..., it remains unclear ... .

l. 59-60. moving speed of an object -> speed of a moving object

l. 81. anticipative time -> anticipative times

l 100, 2 ~ 5 ms \rho is -> 2 ~ 5 ms, \rho is the neural density, and

l 150, dynamical property -> dynamical properties
l 151, of adding the -> of adding

fig 2 B, units of v_int?

l 208, traveling wave -> traveling waves

l 215, the simulation -> simulation
l 215, that at -> that in
l 216, the Gaussian shape -> Gaussian shape

l. 232, is dominated -> are dominated
l. 258, effect on -> effect of

l. 304-306, which gives to (twice) -> gives rise to?
l. 320-321, SFA holds these appealing -> exhibits these properties.
l. 354, Prefect -> Perfect

l. 360, the condition that ... -> the approximation
l. 361, We carry -> We carried
l. 265, of stimulus speed -> of stimulus speeds


throughout the Ms.: holding a traveling wave -> contains a traveling wave / can exhibit traveling waves?

The paper is well written and seems interesting, although I find it hard to judge the novelty of the work. A comparison with experimental data would make the paper more interesting and relevant.

SUMMARY

This was an interesting treatment of tracking dynamics using continuous attractor neural networks. In brief, the authors show that spike frequency adaptation is a sufficient condition to induce travelling waves (moving bump attractors). These provide a plausible explanation for anticipatory dynamics when entrained by exogenous input. The notion is that these dynamics may provide a solution to sensorimotor delay problems in the real brain.

COMMENTS TO AUTHORS

I enjoyed reading this clearly described and well-motivated treatment of continuous attractor networks (and wave equations) in the context of anticipatory dynamics in neuronal networks. I have a few minor points that might improve the presentation of your ideas.

1) It might be nice to refer to predictive coding as a popular framework for understanding active vision and inference that necessarily rests upon some form of prediction and, implicitly, anticipation in the context of oculomotor delays. In this context, your contribution provides a nice and biologically plausible way of finessing the oculomotor delay problem that could – in principle – be absorbed into an inference scheme using Bayesian filtering?

2) On line 53, replace “these supporting evidence” with “this supporting evidence”.

3) On line 101, I would say “which is generally assumed to have the following form”:

4) It would be nice to supplement Figure 1 with the difference between the symmetric and asymmetric interaction kernels (Equations 4 and 5) to give the reader a fuller intuition about the asymmetry.

5) In section 2.3, there may be an important difference between the asymmetric neural networks and the “symmetry breaking” afforded by spike frequency adaptation. I am not sure but the direction of the travelling wave in the asymmetric formulation is pre-determined by the asymmetry in the spatial interaction kernel. If this is the case, then you could say something like:

“Note that both SFA and asymmetric coupling destroy the stability of stationary bump attractors; however, SFA admits moving solutions in either direction. This should be contrasted with the asymmetric coupling implementation; in which the direction of motion is determined by the form of the asymmetry. This could be potentially important in terms of tracking behaviour - in which targets can move in either direction.”

6) I would move Figure 2 to after you have introduced Equation 13 and have defined intrinsic velocity.

7) In Equation 12, I would either replace u with phi or phi with u in the line below to make it clear that these are both the principal eigenmodes.

8) On line 305, replace “gives to” with “gives”.

I hope that these comments help should any revision be required.

This was a compelling demonstration of the utility of spike frequency adaptation in neural field models of travelling waves – that may play a useful role in anticipating or tracking.

The paper uses spike frequency adaptation (SFA) to implement anticipative tracking in a Continuous Attractor Neural Network (CANN).
Anticipative tracking is interesting because the neural delays in actually perceiving a moving object mean that all our perceptions lag reality.
Here the authors show that SFA allows for anticipative tracking over a wide range of velocities

The paper is clearly written, to my knowledge the results are novel, interesting, and the performance seems good.
I recommend including the paper in the proceedings.

The paper is clearly written, to my knowledge the results are novel, interesting, and the performance seems good.

To Reviewer_23:

We acknowledge the valuable comments of the reviewer.

We would like to clarify the novelty of our paper. The previous study [12] only showed that a continuous attractor neural network (CANN) with SFA can retain self-sustained traveling waves (“self-sustained” means that the bump moves spontaneously without relying on external drive), but [12] did not explore the network’s response to external moving inputs, i.e., the tracking behavior of the network. In this work, we elucidate that there exists a close link between the intrinsic mobility of a network (measured by the speed of the traveling wave the network can hold) and its tracking behavior to external moving input, that is, the interplay between the intrinsic mobility of the network and the speed of the external drive determines whether the network tracking is anticipative or lagging with respect to the moving input. Furthermore, we show that by regulating the amplitude of SFA, the CANN can achieve, either perfect tracking (zero-lag) or perfect anticipative tracking (a constant leading time), for a wide range of input speed values, agreeing with the experimental data on the head-direction systems. We expect that this study will give us new insight into understanding how the brain compensates for neural delays pervasive in signal transmission and processing.

According to the literature [10,11], SFA can be generated by a number of biophysical mechanisms and its time constant can range from tens of milliseconds to several seconds. On the other hand, the neuronal time constant is of the order of milliseconds. Hence the choice of the relative magnitudes of the $\tau$ and $\tau_v$ are realistic.

Our model actually reproduces well the experimental data for perfect tracking (zero-lag as shown inFig.4B) by substituting experimentally realistic parameters into the equations. Similarly, our model reproduces well the experimental data for perfect anticipative tracking (a roughly constant leading time as shown in Fig.4D).

Minor issues:

Yes, the flash-lag phenomenon is due to the fact that the flash is not moving.

Fig.2B, the unit of $v_int$ is angle (2\pi)/\tau.

We thank the reviewer for correcting typos and grammar. We will modify them accordingly.


To Reviewer_26:

We acknowledge the encouraging comments of the reviewer.

We fully agree that our model can be formulated in the framework of Bayesian inference, that is, the network uses the history of input to predict the future position of a moving object. Specifically, the history of the input is stored in the SFA synaptic current $V(x, t)$, and the parameters involved in the dynamics of $V(x, t)$ encode the priors.

Concerning the difference between the hard-wired asymmetry couplings and SFA, we agree with the reviewer that a CANN with asymmetric couplings can only support traveling wave in one direction; whereas, a CANN with SFA can support both moving directions depending on that of the input. Apart from this, another key advantage of SFA is that the network can achieve a roughly constant leading time for a wide range of input speed values (Fig.4D), a property which is crucial for practical applications. This property is due to the fact that the amplitude of self-inhibition induced by SFA depends on the external drive, which makes the network response to be “plastic”; whereas, for the hard-wired asymmetric coupling, the leading time of the network will vary with the input speed, and hence it is not really useful.

We thank the reviewer for the favorable suggestions on how to improvethe writing. We will modify the paper accordingly.

To Reviewer_37:

We acknowledge the encouraging comments of the reviewer.

Here, we would like to emphasize a little more on the novelty of this work. The key contribution of this study is the unveiling of a close link between the intrinsic mobility of a neural circuit and the tracking performance of the neural system, elucidating that the former underlies the condition for the neural system to track a moving input anticipatively (in this sense the experimentally observed traveling wave may be regarded as a by-product of the neural circuit being wired in a way to retain sufficient mobility). This close relationship between the intrinsic dynamics of a system and its response property to external inputs is a common theme in many physical systems besides neural systems. Similar relationships between the stability of an equilibrium state and its response to external inputs are expressed as the fluctuation-response theorem, such as the Einstein-Smoluchowski relation on Brownian motion. Therefore, we expect that the interplay between the mobility of localized neural states and the tracking response time is generally applicable to an entire family of neural systems; SFA, together with STD and NFC cited in the concluding section, are just a few examples. Moreover, we expect that the delay compensation strategy proposed in this study may be applied to motion control in artificial dynamical systems.