# An Attractor Model of the Temporal Dynamics of the Songbird's Premotor Nucleus

Fjola Hyseni, Nicolas P. Rougier, Arthur Leblois

► To cite this version:

Fjola Hyseni, Nicolas P. Rougier, Arthur Leblois. An Attractor Model of the Temporal Dynamics of the Songbird's Premotor Nucleus. CNS 2023 - 32nd Annual Computational Neuroscience Meeting, Jun 2023, Leipzig, Germany. . [hal-04444204](https://hal.science/hal-04444204v1)

HAL Id: hal-04444204

https://hal.science/hal-04444204v1

Submitted on 7 Feb 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# An Attractor Model of the Temporal Dynamics of the Songbird's Premotor Nucleus

<figure><img src="image_3.png" /></figure>

université
de BORDEAUX

<figure><img src="image_4.png" /></figure>

FJOLA HYSENI

NICOLAS ROUGIER

ARTHUR LEBLOIS

INSTITUTE OF NEURODEGENERATIVE
DISEASES
BORDEAUX UNIVERSITÉ - CNRS - INRIA

fjola.hyseni@u-bordeaux.fr

## Introduction

<figure><img src="image_5.png" /></figure>

Temporal control is crucial for the generation of coordinated movements during sensorimotor tasks.

<figure><img src="image_6.png" /></figure>

What do songbirds tell us about timing?

In male zebra finches, the premotor nucleus HVC (proper name) is responsible for the precise control of song tempo, becoming as such an ideal model for action timing.

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

How to model motor timing in HVC, in accordance with - electrophysiological (3-6 spikes, 6-10 ms burst) and - behavioral (specificity in adaptive learning) evidence?

Current computational models of HVC rely on synfire chains. We propose attractor models as an alternative.

## 2 Reproducing HVC<sub>RA</sub> dynamics

Rate Based Model:

$$ \frac{d}{dt}m(x,t) = -m(x,t) + G[I_{\text{ext}}(x,t) + I_{\text{syn}}(x,t) - T + \sqrt{\tau_n}\sigma_n\eta(x,t)]^* $$

t=25ms

t=100ms

t=175ms

t=250ms

Mean Firing Rate across Time, $m(t, t)$

<figure><img src="image_10.png" /></figure>

## 1 Modelling motor timing in songbirds: Synfire Chain vs. Ring Attractor.

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

1. We implement a synfire chain with adaptive leaky integrate and fire neurons with synaptic delays and show that the weights between layers can be modified only within a narrow range, the width of which decreases as we increase the number of neurons per layer.

2. An alternative hypothesis is that the gradual propagation of an activity bump is driven in HVC by attractor dynamics. In particular, a linear attractor, also referred to as ring attractor, can drive a drifting activity bump with robust and resilient properties thanks to recurrent connections.

$$ W(x-x') = W_0 + W_2 \frac{1}{\sigma\sqrt{2\pi}} e^{-(\frac{|x-x'|}{\sigma})^2} $$

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

Exponential Integrate & Fire neuron with adaptation (AdEx) and conduction delays:

$$ C \frac{dV}{dt} = -g_L(V - E_L) + g_L \Delta T \exp\left(-\frac{V - V_T}{\Delta T}\right) + I_{\text{ext}} - w + I_{\text{syn}} + \sqrt{\tau_n} \sigma \eta_k(t)^* $$

<figure><img src="image_15.png" /></figure>

Hahlo遳er et al., Nature 2002

t=15ms

t=40ms

t=65ms

t=90ms

<figure><img src="image_16.png" /></figure>

Mean Firing Rate across Time, $m(t, t)$

<figure><img src="image_17.png" /></figure>

- Using the rate based model, only a short duration bump profile can be generated, but a small membrane time constant (1 ms) is necessary to reproduce the 10 ms burst durations. With the spiking (SNN) model, we acquire results similar to the ones observed in the rate model, but are able to generate experimentally-observed neuronal dynamics with bioplausible parameters.

## 3 Behavioral Adaptation with Reinforcement Learning

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

Relying on a reward covariance rule (Williams, 1992), we show that the duration of a syllable can be modified in response to a perturbed reward profile, as implemented in a widely used lab conditioning paradigm.

$$ \Delta W_{ij} = \gamma R e_{ij} dt $$

$$ e_{ij} = \int_{0}^{t} \frac{dt'}{\tau_e} e^{-(t-t')/\tau_e} \eta_k(t') r_j(t') $$

Consistent with behavioral data, the change in duration is specific to the target syllable. We see no interference with adjacent or non-adjacent syllables, both in the case of targeting for duration increase and decrease.

<figure><img src="image_22.png" /></figure>

## 4 Discussion and Conclusion

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

Song timing: HVC (HVC<sub>RA</sub> neurons)

<figure><img src="image_26.png" /></figure>

Synfire Chain (not robust)
Ring Attractor (robust)

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

AdEx with delays; 3-6 spikes, 6-10 ms
duration (~HVC<sub>RA</sub> neurons)

<figure><img src="image_29.png" /></figure>

Specific Adaptive Learning
(~ behavioral experiments)

## References

[1] Jin et al., J Comput Neurosci, 2007: Intrinsic bursting enhances the robustness of a neural network model of sequence generation by avian brain area HVC.

[2] Hansel & Sompolinsky, 1998: Modeling feature selectivity in local cortical circuits.

[3] Egger et al., Cell, 2020: Local axonal conduction shapes the spatiotemporal properties of neural sequences.

[4] Pehlevan et al., Nat. Comm., 2018: Flexibility in motor timing constrains the topology and dynamics of pattern generator circuits.

[5] Buonomano & Laje, 2010: Population clocks: motor timing with neural dynamics.

[6] Williams, Mach. Learn., 1992: Simple statistical gradient-following algorithms for connectionist reinforcement learning.

<footnote>* $m(x, t)$: mean firing rate of neuron $x$ at time $t$. G: semi-linear gain function, $\tau$: membrane time constant, $I_{\text{ext}}$: external input, $I_{\text{syn}}$: synaptic input, $T$: threshold, $\sqrt{\tau_n}\sigma_n\eta(x, t)$: zero mean Gaussian white noise.</footnote>

<footnote>** V: membrane potential, C: membrane capacitance, g_L: leak conductance, E_L: leak reversal potential, w: adaptation variable, V_T: threshold, $\Delta T$: slope factor.</footnote>