Is the Hippocampus a Helmholtz Machine

[NorbertZheng]

Overview

A consensus is emerging that the Hippocampal formation functions by combining external sensory data (LEC) with internal position encodings (MEC) to form a ‘cognitive map’ (HPC). However, existing models are biologically implausible in their architecture and/or learning rules. Here I will discuss some preliminary work casting hippocampus as a hierarchical “Helmholtz machine” which learns hidden structure in the data by minimising predictive-coding style errors. This is more plausible (and less complicated) than it sounds; the learning rules are local and Hebbian, each hierarchical layer maps onto a known layer in the hippocampal formation and the forward/backward sweeps are explained by neural oscillations in what is a biological approximation to back prop. The model accounts well for ability of mammals to do “path integration”, and may shed light on compressed position encodings (grids?), as well as the phenomena of remapping and theta sweeps.

[NorbertZheng]

Related Reference

1. McNaughton B L, Battaglia F P, Jensen O, et al. Path integration and the neural basis of the 'cognitive map'.
2. Dordek Y, Soudry D, Meir R, et al. Extracting grid cell characteristics from place cell inputs using non-negative principal component analysis.
3. Stachenfeld K L, Botvinick M M, Gershman S J. The hippocampus as a predictive map.
4. O'Keefe J, Nadel L, Keightley S, et al. Fornix lesions selectively abolish place learning in the rat.
5. Bush D, Barry C, Burgess N. What do grid cells contribute to place cell firing.
6. Whittington J C R, Warren J, Behrens T E J. Relating transformers to models and neural representations of the hippocampal formation.
7. Sanders H, Rennó-Costa C, Idiart M, et al. Grid cells and place cells: an integrated view of their navigational and memory function.
8. Hasselmo M E, Stern C E. Theta rhythm and the encoding and retrieval of space and time.
9. Bredenberg C, Simoncelli E, Savin C. Learning efficient task-dependent representations with synaptic plasticity.
10. Banino A, Barry C, Uria B, et al. Vector-based navigation using grid-like representations in artificial agents.
11. Sorscher B, Mel G C, Ocko S A, et al. A unified theory for the computational and mechanistic origins of grid cells.
12. Whittington J C R, Muller T H, Mark S, et al. The Tolman-Eichenbaum machine: unifying space and relational memory through generalization in the hippocampal formation.
13. Hasselmo M E, Bodelón C, Wyble B P. A proposed function for hippocampal theta rhythm: separate phases of encoding and retrieval enhance reversal of prior learning.
14. Bittner K C, Grienberger C, Vaidya S P, et al. Conjunctive input processing drives feature selectivity in hippocampal CA1 neurons.
15. Brankačk J, Stewart M, Fox S E. Current source density analysis of the hippocampal theta rhythm: associated sustained potentials and candidate synaptic generators.
16. Mizuseki K, Sirota A, Pastalkova E, et al. Theta oscillations provide temporal windows for local circuit computation in the entorhinal-hippocampal loop.
17. Clopath C, Büsing L, Vasilaki E, et al. Connectivity reflects coding: a model of voltage-based STDP with homeostasis.
18. Urbanczik R, Senn W. Learning by the dendritic prediction of somatic spiking.
19. Brea J, Gerstner W. Does computational neuroscience need new synaptic learning paradigms.
20. Asabuki T, Fukai T. Somatodendritic consistency check for temporal feature segmentation.

[NorbertZheng]

Motivation: HPC-EC connectivity

... and why this matches that of a generative model, specifically the Helmholtz machine.

Hippocampus and Entorhinal cortex have bi-directional connectivity

Medial entorhinal cortex (grid cells): the "cognitive graph".

• Location of path integration [1].
• Compressed representation of space [2,3].

Hippocampus (place cells): the "cognitive map".

• Binding sensory to internal representation [4,5].

Lateral entorhinal cortex: sensory data, landmarks.

[NorbertZheng]

Claim:

• This looks like a generative model.
• We lack a bio-plausible instantiation (information flow and learning).

[NorbertZheng]

Hippocampal activity is dominated by theta oscillations

[NorbertZheng]

Proposal:

• HPC-EC is a generative [6] model similar to Helmholtz machine.
• Sleep-wake cycles are controlled by theta oscillations.
• The learning rule is local and Hebbian.

[NorbertZheng]

Learning latent representations (Helmholtz machine)

Learn latent structure $p{\theta}(\mathbf{z}|\mathbf{x})$ in a dataset $\{\mathbf{x}\}$ by simultaneously learning a generative model, $p{\phi}(\mathbf{x}|\mathbf{z})$.

When (and only when) your generative model can accurately predict the $\mathbf{x}$ from $\mathbf{z}$, you have learnt "good" latent variables.

[NorbertZheng]

Can extend this to learn latents with temporal structure.

[NorbertZheng]

Theta:

• gates information flow, controlling whether system
  • infers location from sensory input,
  • generates location from path integration (like [7]).
• controls plasticity (like [8]). Rapid oscillations, flipping between inference and generation trains hidden representations analogous to wake-sleep cycles of a Helmholtz machine (like [9]).

This would explain:

• Place cells without grid cells: only inference.
• Theta sweeps.
• Accurate navigation in noisy environments: integrate inference and generation.

[NorbertZheng]

There exists some evidence for this ...

Computational models:

• Tolman-Eichenbaum Machine [12].

  • Local learning rules: x
  • Theta oscillations: x
  • Generative model: √

  

• SPEAR (Seperate Phases of Encoding And Retrieval) [13].

  • Local learning rules: √
  • Theta oscillations: √
  • Generative model: x

  

• Mind Travel [7].

  • Local learning rules: x
  • Theta oscillations: √
  • Generative model: √

  

Electrophysiological evidence:

• Bittner et. al. 2015.
• Brankačk et. al. 1993.
• Mizuseki et. al. 2009.

[NorbertZheng]

Proposal: Local oscillatory learning model

aka: what does this learning rule actually look like?

Learning rule just is dendritic prediction of somatic activity

$$ \begin{aligned} \Delta \vec{w}&=\eta\left[\mathbf{U}(t)-\mathbf{V}(t)\right]\cdot\vec{I}(t)\ \Delta \vec{w}&=\eta\left[\phi(\mathbf{U}(t))-\phi(\mathbf{V}(t))\right]\cdot\phi'(\mathbf{V}(t))\cdot\vec{I}(t)\ \end{aligned} $$

The above is "Voltage dependent" Hebbian learning [1,2,3,4].

[NorbertZheng]

Theta gates entry of dendritic voltage into soma

• Early - $\theta$:

  • Bottom-up "basal" dendrite drive soma.
  • Apical dendrites adjust weights to match.

  

• Late - $\theta$:

  • Top-down "apical" dendrites drive soma.
  • Basal dendtrites adjust weights to match.

  

[NorbertZheng]

The architecture of network

[NorbertZheng]

Results

Spatial information compression

[NorbertZheng]

Path integration

[NorbertZheng]

[NorbertZheng]

[NorbertZheng]

[NorbertZheng]

Conclusions