Lecture 3: Neurorobotic models of spacial cognition

Teacher: Anfelo Arleo

Introduction

Stimulus ⟶ Response

  • Experimental approach: tuning curves: plotting activity (spikes/sec) as a function of the stimulus (ex: the angle)

  • How to extrapolate? ⇒ Model of the system

    • Model of the system: set of equations underlying the $S-R$ relation and provide testable predictions

  • Parallel aim: reduce the complexity of the model

Equations:

  1. analytical solutions, but very rare
  2. simulations (95% of the time)
  3. implement the model on a real robot, and make sure it works in practice, in the real world (not just the simulated one)

Computational neuroscientist: not necessarily a good programmer, nowadays, we have a myriad of librairies/tools to help with the simulation

Hardness ⟶ in the model

Complex system ⟶ Experimental protocol ⟶ Experimental data

Then compare with the mathematical model ⟶ simulations

NB: the aim is to make the model good at predicting, not just describing

Neural activity and neural coding

Stimulus ⟶ Encoding ⟶ Response

Encoding: the process that encodes the stimulus in the brain

What we want is to be able to decode: going from brain to inferring the stimulus

Nowadays: we can record activity of $≃ 200$ neurons at the same time (with one electrode)

Decoding

Two approaches:

1. Firing rate decoding approaches ⟶ been done for ages

Problem with computing the average, variance, moments ⟹ we lose the time-related information

Ex: 4 spikes fired at different times carry different information, a priori!

1968 experiment ⟶ neurons in the $V_1$ area are sensitive to orientation

You might

  • get all the tuning curve, for each neuron, one by one
  • use the “winners take all” approach ⟶ with one electrode, you record the simultaneous activity of $≃ 200$ neurons, then plot a “combined tuning curve”, then keep only the most firing ones

  • “population vector” decoding ⟶ weighted sum (cooperative)

    Ex: if you do the average of the vector responses (in the polar plane: argument/angle = preferred direction, norm = firing rate), then you can guess the direction the agent is focusing on ⇒ neuroprostheses

    Neural feedback: you get the feedback, to adjust your thinking

    Now we even do the same for vision, to overcome blindness! Intead of deconding, we encode (ex: light from the eyes), then send the information to the brain

Firing rate neuronal model

For each neuron $i$

  • $V_i(t)$: membrane potential
  • $I_i(t)$: synaptic input
  • $f(V_i)$: transfer function
  • $r_i(t)$: firing rate
\[τ_i \frac{dV_i}{dt}(t) = - V_i(t) + I_i(t)\]

where

\[I_i(t) = \sum\limits_{ j } w_{i, j} r_iε\]

Integrate & Fire: more advanced model, takes timing into account (up to 3 differential equations per neuron)

Hodgkin & Huxley: even more complex, up to 25 equations per neuron

Types of learning

  • Supervized: at first, not thought to happen at a neuronal level, but it has been shown recently to happen in the cerebellum

  • Reinforcement: happens all the time

  • Unsupervized: clustering, happens a lot

Hebbian model: «Neuron that fire together wire together» ⟹ you can get a simple model of memory (associative memory)

Spatial cognition

Ex: put a mouse in a pool, where there is

  • a visible platform ⟶ the mouse goes directly to the platform
  • an invisible platform ⟶ takes more time to find the platform, then learn faster and faster

Spatial cognition: neuron may fire only on specific positions/locations in space.

Neural basis of spatial cognition

Experiment: identify neurons that fire in

  • specific locations ⟶ then you can reconstruct a map of movement of the animal based on these neurons activity

  • specific directions ⟶ these neurons act as a compass, independently of the position of the animal

Grid cells: discovered in 2005 ⟶ neuron maps a “grid” hardcoded in the brain

And before experimental evidence in 2005, the existence of these grids was already predicted by many models (path integration ⟶ when you keep your closed when moving ⟹ path integration)

Border cells: neurons that respond to the agent’s distance to walls

  • Allothetic sensory inputs: come from the external environment
  • Idiothetic sensory inputs: come from the agent’s body

Environemental landmarks: very precise, you have visual cues to find your way

Path integration: far less precise, you quickly lose track of where you came from

Berthoze: vection illusion (when you think your going backwards in train at the beginning when you see another going in the opposite direction)

Experiment: have a planetarium with light dots confuse the animal’s direction, which thinks the lights are not moving and it is.

⇒ visual cues overtook motor cues

Use Hebbian learning/STDP to make a robot move in space ⟹ combine visual and motor cues to strike a balance and prevent ambiguity

Books:

  • Neural Dynamics
  • Theoretical Neuroscience
  • Spikes

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