Combining sampling and attractor dynamics in spiking models of head direction systems.
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The authors' code
Jupyter notebook · 57 lines · 1.9 KB · GPL-3.0
- # %% [markdown]
- # # Combining Sampling Methods with Attractor Dynamics in Spiking Models of Head-Direction Systems
- # ## by Vojko Pjanovic, Jacob Zavatone-Veth, Paul Masset, Sander Keemink & Michele Nardin.
- #
- # Code written by Vojko Pjanovic and Michele Nardin.
- #
- # This notebook includes the code for Fig. 1 panels B and C.
- # %% [markdown]
- # ## Panel B:
- # #### For increasing noise levels on the encoding of the angular velocity, the posterior distribution 𝑃 (𝜔𝑡 |𝝈𝑡 ) will show more and more uncertainty.
- #
- # The posterior over Poisson spikes is a Gumbel distribution:
- # $P(\omega_t | \sigma_t) \propto e^{-e^{\beta \omega_t}}e^{\beta \omega_t \sigma_t}$
- # %%
- import numpy as np
- import matplotlib.pyplot as plt
- # plot pdf of gumbel distribution
- omega = 1
- np.random.seed(0)
- for beta in [0.5, 1, 3]:
- kernel = np.random.poisson(np.exp(beta*omega))
- x = np.linspace(-3,5,1000)
- plt.figure(figsize=(2,2))
- y = np.exp(beta*kernel*x - np.exp(beta*x))
- plt.plot(x,y)
- plt.axis('off')
- plt.title(f'beta={beta}, kernel={kernel}')
- plt.show()
- # %% [markdown]
- #
- # ## Panel C:
- # #### Estimating true density and statistics through sampling (colorful arrows: samples, grey line: samples histogram, blue line: true distribution).
- # %%
- # sample 20 samples from gumbel distribution with beta = 1
- # then plot these samples as small arrows, each of different color, on the x axis
- np.random.seed(13)
- samples = np.random.gumbel(0,1,12)
- plt.figure(figsize=(2,2))
- # plot pdf first
- x = np.linspace(-3,5,1000)
- y = np.exp(-x - np.exp(-x))
- plt.plot(x,y,label='true density')
- plt.plot(np.linspace(-3,5,1000),np.zeros(1000),'k')
- for i,sample in enumerate(samples):
- plt.arrow(sample,-0.11,0,0.1,color=plt.cm.viridis(i/len(samples)),head_width=0.1,head_length=0.01,label='sample '+str(i))
- plt.axis('off')
- # plot the hist
- plt.hist(samples,bins=np.linspace(-3,5,8),density=True,color='k',histtype='step',label='samples hist.',alpha=0.5)
- plt.show()
Figure1_v2.ipynb at commit 45a30d4, under GPL-3.0 · at the source
Overview
- Janelia Research Campus, Howard Hughes Medical Institute, Ashburn, Virginia, United States of America
- Department of Machine Learning and Neural Computing, Donders Institute for Brain, Cognition and Behaviour, Radboud University, Nijmegen, Netherlands
- Society of Fellows and Center for Brain Science, Harvard University, Cambridge, Massachusetts, United States of America
- Department of Psychology, McGill University, Montréal, Québec, Canada
- Mila - Quebec Artificial Intelligence Institute, Montréal, Québec, Canada
Abstract
Neural populations can maintain stable representations of navigation-related variables while integrating uncertain sensory signals. Experimental evidence showed that the precision of head-direction (HD) representations in flies and mice depends on the reliability of sensory cues, highlighting the influence of input uncertainty in attractor-based neural circuits. How do neural dynamics maintain stability while computing under uncertainty? Here, we propose a spiking neural network that unifies two principles — stability through attraction and uncertainty through fluctuation — and reinterpret the HD circuit as an uncertainty-aware integrator rather than a deterministic compass. Specifically, the network uses sampling-based probabilistic inference, where a neural population represents input uncertainty by rapidly fluctuating among likely hypotheses about the world while preserving a stable representation of head direction along an attractor manifold. This formulation suggests why a classical HD “bump” becomes less precise, namely due to rapid fluctuations, reflecting the uncertainty in angular velocity inputs. Our implementation yields experimentally testable predictions: correlated subthreshold voltage fluctuations, multi-timescale nonlinear interaction patterns, and characteristic statistics of bump movement. By combining probabilistic inference with attractor dynamics within one single circuit, our framework suggests how neural populations across species can represent an estimate and its uncertainty through fluctuations while maintaining stability, which could be a general principle for uncertainty-aware computation in noisy biological systems.
Reproduced under the paper's license (CC BY), from the paper cited above.
Repository
Its files are read in the Code ↔ Paper reader above.
michnard/sampl_attract_HD
45a30d440031ce4e728226ee64a2ebb7d79a6de8, 20 January 2026Availability: 1 check, the latest on 27 September 2026: the link answers
- 27 September 2026: the link answers
7 files
- Figure1_v2.ipynb, Jupyter, 57 lines
- Figure2_v3.ipynb, Jupyter, 174 lines
- Figure3_v2.ipynb, Jupyter, 196 lines
- Figure4_v2.ipynb, Jupyter, 256 lines
- funcs.py, Python, 56 lines
- LICENSE, License, 674 lines
- README.md, Text, 9 lines
The paper's code and data availability statement is in the Data section.
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Data Availability
The code to generate the simulations and reproduce the analyses in the main figures is available at https://
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Version 1, 27 September 2026: the first record
Recorded: type, language, journal, volume, issue, pages, dates, 5 authors, 10 MeSH terms, 6 funders, 79 references.
Cite
This paper
Pjanovic, V., Zavatone-Veth, J. A., Masset, P., Keemink, S. W., & Nardin, M. (2026). Combining sampling and attractor dynamics in spiking models of head direction systems. PLoS computational biology, 22(7), e1014577. https://
BibTeX
@article{pjanovic2026com
author = {Pjanovic, Vojko and Zavatone-Veth, Jacob A. and Masset, Paul and Keemink, Sander W. and Nardin, Michele},
title = {{Combining sampling and attractor dynamics in spiking models of head direction systems}},
journal = {PLoS computational biology},
year = {2026},
month = jul,
volume = {22},
number = {7},
pages = {e1014577},
publisher = {PLOS},
issn = {1553-734X},
doi = {10.1371/
url = {https://
pmid = {42531351},
pmcid = {PMC13450861}
}
RIS
TY - JOUR
AU - Pjanovic, Vojko
AU - Zavatone-Veth, Jacob A.
AU - Masset, Paul
AU - Keemink, Sander W.
AU - Nardin, Michele
TI - Combining sampling and attractor dynamics in spiking models of head direction systems
T2 - PLoS computational biology
J2 - PLoS Comput Biol
PY - 2026
DA - 2026/
VL - 22
IS - 7
SP - e1014577
SN - 1553-734X
PB - PLOS
DO - 10.1371/
UR - https://
LA - en
ER -
CSL-JSON
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