Learning Bayes-optimal
dendritic opinion pooling
Jakob Jordan
Department of Physiology, University of Bern, Switzerland
25.08.2022, Giessbach meeting, Giessbach, Switzerland
Jordan, J., Sacramento, J., Wybo, W. A., Petrovici, M. A., & Senn, W. (2021).
Learning Bayes-optimal dendritic opinion pooling. arXiv preprint arXiv:2104.13238.
(High-level) neuronal representations reflect contributions from multiple modalities
How to combine information from uncertain sources?
\bm z
visual
auditory
olfactory
\sigma_\text{v}^2
\sigma_\text{a}^2
\sigma_\text{o}^2
"uncertainty"
How to estimate the uncertainty of each source?
Can single cortical neurons learn to optimally integrate information from uncertain sources?
Can single model neurons with plausible dynamics learn to optimally integrate information from uncertain sources?
An observation
Bayes-optimal inference
"Slow" membrane potentials
mean
precision
Membrane potential dynamics from noisy gradient ascent
\begin{array}{rl}
p(u_\text{s}|W,r)
\end{array}
\begin{array}{rl}
\dot u_\text{s} \propto \frac{\partial}{\partial u_\text{s}} \log p(u_\text{s}| W,r) + \xi
\end{array}
\mathbb{E}[u_\text{s}] = \bar E_\text{s}
Membrane potential mean
= mean of the posterior
Membrane potential variance
= inverse precision of the posterior
\text{Var}[u_\text{s}] = \frac{1}{\bar g_\text{s}}
Somatic membrane potential distribution
Somatic membrane potential dynamics
Synaptic plasticity from matching target distributions
\mathbb{E} \left[ \Delta \mu^{\text{E/I}} \right] \propto (\mu^* - \bar E_\text{s})
\mathbb{E} \left[ \Delta \sigma^2 \right] \propto \frac{1}{\bar g_\text{s}} - \mathbb{E} \left[ (u_\text{s}^* - \bar E_\text{s})^2 \right]
Prediction for experiments:
Synaptic plasticity modifies excitatory/inhibitory synapses
- in approx. opposite directions to match the mean
- in identical directions to match the variance
\dot W_d^\text{E/I} \propto [ \; \Delta \mu^\text{E/I} \, + \, \Delta \sigma^2 \; ] \, r
target
actual
Learning Bayes-optimal inference of orientations from multimodal stimuli
The trained model approximates ideal observers
and reproduces psychophysical signatures of experimental data
[Nikbakht et al., 2018]
Cross-modal suppression as
reliability-weighted opinion pooling
- low stimulus intensities: firing rate is enhanced
- high stimulus intensities: firing rate is suppressed
- prediction for experiments: strength of suppression depends on relative reliabilities of the two modalities
[Ohshiro et al., 2017]
Summary
Single neurons with conductance-based synapses can learn to be optimal cue integrators.
Jordan, J., Sacramento, J., Wybo, W. A., Petrovici, M. A., & Senn, W. (2021).
Learning Bayes-optimal dendritic opinion pooling. arXiv preprint arXiv:2104.13238.
- Membrane-potential dynamics compute maximum-a-posteriori estimates.
- Membrane conductances represent reliabilities of these estimates.
- Synaptic plasticity allows neurons to match target membrane potential distributions.
Neurons with conductance-based synapses
naturally implement probabilistic cue integration
Membrane potential dynamics from noisy gradient ascent
\begin{array}{rl}
p(u_\text{s}|W,r) =& \frac{1}{Z'} \prod_{d=0}^D p_d(u_\text{s}|W_d,r) \\
=& \frac{1}{Z} e^{-\frac{\bar g_\text{s}}{2}\left( u_\text{s} - \bar E_\text{s}\right)^2}
\end{array}
\begin{array}{rl}
C \dot u_\text{s} =& \frac{\partial}{\partial u_\text{s}} \log p(u_\text{s}| W,r) + \xi \\
=& \sum_{d=0}^D \left( g_d^\text{L} (E^\text{L} - u_\text{s}) + g_d^\text{E} (E^\text{E} - u_\text{s}) + g_d^\text{I} (E^\text{I} - u_\text{s}) \right) + \xi
\end{array}
\mathbb{E}[u_\text{s}] = \bar E_\text{s}
Time-averaged membrane potentials
= mean of the posterior
Membrane potential variance
= inverse precision of the posterior
\text{Var}[u_\text{s}] = \frac{1}{\bar g_\text{s}}
Synaptic plasticity from stochastic gradient ascent
\begin{array}{rl}
\dot W_d^\text{E/I} \propto \frac{\partial}{\partial W_d^\text{E/I}} \log p(u_\text{s}^*|W,r) \\
\end{array}
\Delta \mu^{\text{E/I}} \propto (u_\text{s}^* - \bar E_\text{s}) \left( E^\text{E/I} - \bar E_\text{s} \right) \\
\Delta \sigma^2 \propto \frac{1}{2} \left( \frac{1}{\bar g_\text{s}} - (u_\text{s}^* - \bar E_\text{s})^2 \right)
Prediction for experiments:
Synaptic plasticity modifies excitatory/inhibitory synapses
- in approx. opposite directions to match the mean
- in identical directions to match the variance
\propto [ \; \Delta \mu^\text{E/I} \, + \, \Delta \sigma^2 \; ] \, r
\(u_\text{s}^*\): sample from target distribution \(p^*(u_\text{s})\)
target
actual
Synaptic plasticity performs
error-correction and reliability matching
Learning Bayes-optimal dendritic opinion pooling
By jakobj
