Cortical Organization

Keeley Erhardt and Daniel Fitzgerald

4 PAPERS- Slowness and sparseness lead to place, head-direction,

and spatial-view cells. Franzius, M., Sprekeler, H., & Wiskott, L. (2007)

- Towards a Mathematical Theory of Cortical Micro-Circuits. Dileep George, Jeff Hawkins. (2009)

- Frequently Asked Questions for: The Atoms of Neural Computation. Gary F. Marcus, Adam H. Marblestone, Thomas L. Dean. (2014)

- Neurodynamics of mental exploration. Hopfield, J. J. (2009)

Paper 2: Slowness and Sparseness Lead to Place, Head-Direction, and Spatial-View CellsGoal: Present and verify mathematical model for the three cell types involved in spatial localization.

Steps: A self-organizing hierarchy of Slow Feature Analysis (SFA) nodes extracts sparse-coded position and orientation.

Place Cells● Discovered 1971 (John O’Keefe, Jonathon Dostrovsky)● Hippocampus CA1 and CA3● Can be orientation specific for linear tracks

Grid Cells● Discovered 1992 (Edvard and May-Britt Moser)● Entohinal Cortex

Head-Direction Cells● Discovered 1984 (James Ranck) ● Located in many brain regions● Head “compas” direction● Independent of head orientation relative to body● Mostly based on integration of vestibular accelerations● 3D orientation for bats

Spatial-View Cells● Discovered by Edmund Rolls (1999)● Located in hippocampus● Fire when looking at a specific place

“Oriospatial” Cell Type Comparison

Terminology● Idiothetic: “internal” sensing

○ proprioception, muscle feedback, vestibular, etc.○ Use for “dead-reckoning” integration

● Allothetic: “external sensing”○ vision, olfactory, touch○ Used for error/drift correction

Mathematical Model● No memory (no path integration, Markov localization)● Learns from raw complex visual input● Environment feature extraction based on “Slowness

Principle” of invariant representations (Slow Feature Analysis)

● Each layers learns slowest features of previous layer● Highest SFA layer forms distributed oriospatial

representation

Slow Feature Analysis● Laurenz Wiskott, 1998● Unsupervised learning to optimize nonlinear scaling

function g(x(t)) for time-dependant training data x(t) giving slowest output signal(s) y(t)

Slow Feature Analysis Cont.● “Slowness” measured by Δ-value: mean-square of time

derivative● Additional constraints that y(t) functions be

uncorrelated (think of it as a SVM for independant slow features)

● Note: different from low-pass filter (slowly-varying signals extracted instantaneously by g).

Model, Cont● Top SFA layer output sparse-coded, resulting in localized

oriospatial codes (cell types).● When body movement is not correlated to head direction,

model learns head-direction or place cells.● When view is fixated on locations during movements, model

learns spatial-view cells.

Methods● Approximate retinal stimuli by textured Virtual Reality

Environment.● Motion generated by bounding brownian motion.● Head direction generated by (un)restricted brownian

motion (can impose 90° constraint relative to body)● “Momentum” term controls smoothness of motion● Learning Rate Adaptation (LRA) downgrades learning rate

during quick head turns from forced body turns.● Also, random fixation points on walls attract view (head

direction) - “spatial view”

Architecture and Training● 3-Layer SFA network (7x63, 2x15, 1x1) ● Lower two layers (visual) clamped for different sets● Top node output sparse-coded with Independent Component

Analysis (ICA) or Competitive Learning (CL)● 100,000 time points training data● Python MDP toolbox

Architecture Cont.

AnalysisMapping oriospatial data to images

● Position, orientation = “configuration” (pose) s● Input view image seen at s is x(s)● Reversible: can uniquely determine s from x● Rat’s behavior modeled as position, velocity probability

densities● ...math…● Higher resolution in low-velocity regions (near walls)● Predicts smaller, oriented place fields near boundaries

Results: Unrestricted head● Left column: Movement

speeds slower (selective to nonlocalized position, head-direction invariant)

● Right column: Rotational speeds slower (selective to head direction, position invariant)

● A,D: Theoretical● B,C: Simulated SFA● C,D: Simulated ICA

Results: Restricted Head● Head-direction

within 90° of body movement

● LRA applied to top SFA layer to compensate for high rotation speeds

● CL replaces ICA● place cells● head-direction cells● More SFA nodes ->

convex

Results: Spatial-ViewNote: no units invariant to position or head-direction, because they are now correlated through fixation

Global head-direction Local head-direction

SFA

ICA

Results: Linear TrackHead-direction collapses to binary value (movement direction, N/S)

Theoretical

SFA

ICA

Conclusion● Same learning mechanism, environment result in different

cell types learned depending on movement statistics.● Equivalent to adjusted temporal learning rates.● Learned place, head-direction, spatial-view, grid cells.● Mathematical model with exact analytical predictions● Imposing limited time window (biologically plausible)

also imposes intrinsic spatial scale, hexagonal vs. rectangular grid cell pattern

● Localization not based on landmarks (object recognition)● Robust to noise, but might be sensitive to sunlight, etc.● Can learn place, head-Direction cells at the same time

with vestibular data included

Paper 2: Towards a Mathematical Theory of Cortical MicrocircuitsGoal: Present Hierarchical Temporal Memory as a model for cortical circuits.

Steps:

● Hierarchical bayesian inference as theoretical framework for cortical computation

● Coincidence detectors, Markov Chains● Anatomically derived organizational constraints● Based on Memory-Prediction Framework (Hawkins)● Map mathematics of HTM to anatomy of cortical columns

Hierarchical Temporal Memory● Neorcortex = tree of node nodes units● Nodes store sequences of spatial patterns● Output strength corresponds to agreement with sequence● Higher level sequences constructed from co-occurrences of

lower-level Markov-chains● Higher levels = larger space, longer time scale patterns● Feed-forward (recognition), feed-back (expectation)● Efficiently models spatio-temporal organization of the

real world

HTM as neocortical algorithm“An area of cortex can be thought of as encoding a set of patterns and sequences in relation to the patterns and sequences in regions hierarchically above and below it. The patterns correspond to the coincidence patterns in an HTM node and the sequences correspond to the Markov chains.”

HTM LearningEach node must

● Memorize coincident patterns in children● Learn Markov Chains (probabilities) of those patterns● Can’t memorize all coincidences - store random subset● Multiple patterns can be active at a time● Temporal proximity -> slowness -> invariance● Bayesian belief propagation for top-down inference

HTM Cont.

Belief propagation● Node receives Degree of Certainty (DoC) over child’s

Markov Chains (read: expected value of input patterns)● Node updates its prob dist over it’s coincident patterns● Node recalculates its own DoCs for its Markov Chains● Node sends DoCs to parents● Node receives parent’s DoCs for this Node’s Markov Chains● Node recalculates prob dist over coincident patterns● Node recalculates DoCs for it’s child nodes● Node sends Markov Chain DoCs to children

Neuronal implementationPropose neurons types exist that

● Detect coincidences● Represent coincidence patterns● Represent Markov Chains● Calculate prob dists over markov chains from prob dist

over coincidences● Calculate prob dists over coincidences from prob dist

over Markov Chains ● Calculate beliefs values by pooling over same coincidence

patterns in different Markov Chains

Neural Implementation ContThese neuron types are mapped to a cortical column model

● Connections within column can mostly be established without learning (supported by developmental neuroscience?)

Experiments and resultsTested well on

● Object Recognition (Caltech-101)● Top DOwn Segmentation (using feedback)● Subjective contour effect (Expectation demonstration)

Paper 3: Faqs for: The Atoms of Neural Computation

Goal: Construct an improved taxonomy and phylogeny of cortical computation

The human cerebral cortex- The cerebral cortex is the brain’s outer layer of neural

tissue- Central to a wide array of cognitive functions

- Vision- Language- Reasoning- Decision-making- Motor control

- Basic logic remains unknown

The human cerebral cortex cont.Prevailing Hypothesis: cortical neurons form a single, massively repeated “canonical” circuit

- Does a single uniform canonical cortical circuit exist?

Experimental Proof For Cortical Uniformity- Series of experiments by Sur and collaborators (Sharma,

Angelucci, & Sur, 2000), based on (Frost & Metin, 1985)- Visual inputs to primary visual cortex (V1) were rerouted

to the primary auditory cortex (A1) which was capable of processing visual stimuli

- Often taken to imply a “uniform” cortical substrate

Caveats to sur’s experimental results1. Similar results have only been demonstrated within

primary sensory cortices2. The “rewired” auditory complex sortex still retains some

of its intrinsic properties and the resulting “visual” system is not without defects

3. The areas were not directly “rewired”

Doubts regarding a uniform architecture- Can a uniform architecture capture the diversity of

cortical function in simple mammals?- Can it capture characteristically human processes such as

language and abstract thinking?

Canonical circuit + Analogous ai (e.g. deep learning nets)Effective:

- Certain pattern classification tasks (i.e. speech and image recognition)

Less Effective:

- Reasoning- Natural language understanding

A new alternative modelCortex consists of a diverse set of computationally distinct building blocks that implement a broad range of elementary, reusable computations

- Reusable computational primitives versus a single canonical circuit

- Computational primitives: elementary units of processing similar to sets of basic instructions in a microprocessor

Exploratory example taxonomy of cortical computation

Rapid perceptual classification

Complex spatiotemporal pattern recognition

Learning efficient coding of inputs

Working memory

Decision making

Potential algorithmic/ representational realization(s)Computation

Potential neuronal implementation(s)

Putative brain location(s)

Receptive fields, pooling and local contrast normalization

Bayesian belief propagation

Sparse coding

Continuous or discrete attractor states in networks

Reinforcement learning of action-selection policies in PFC/BG system AND Winner-take-all networks

Hierarchies of simple and complex cells

Feedforward and feedback pathways in cortical hierarchy

Thresholding and local competition

Persistent activity in recurrent networks

Recurrent networks coupled via lateral inhibition

Visual system

Sensory hierarchies

Sensory and other systems

Prefrontal cortex

Prefrontal cortex and basal ganglia

Exploratory example taxonomy of cortical computation cont.

Routing of information flow

Gain control

Sequencing of events over time

Representation and transformation of variables

Variable binding

Potential algorithmic/ representational realization(s)Computation

Potential neuronal implementation(s)

Putative brain location(s)

Context-dependent tuning of activity in recurrent network dynamics AND Shifter circuits AND Oscillatory coupling

Divisive normalization

Feed-forward cascades AND Serial working memories

Population coding

Indirection AND Dynamically partitionable autoassociative networks

Recurrent networks implementing line attractors and selection vectors

Frequency filtering via feedforward inhibitionSynfire chains

Ordinal serial encoding through variable binding

Time-varying firing rates of neurons and generalizations to higher dimensions

Common across many cortical areas

Common across many cortical areasLanguage and motor areas/Prefrontal cortex

Motor cortex and higher cortical areas

Prefrontal cortex/basal ganglia loops/Higher cortical areas

BIOLOGICAL EVIDENCE FOR CORTICAL DIVERSITY- Coarse level

- Cytoarchitectonic (cellular composition of the body’s tissues) differences

- Differences in the distribution of different types of interneurons between areas

- Canonical structural features present in select parts of the cortex

- Local microcircuitry- E.g. Variance in synaptic connectivity and synaptic properties

- Differences between gene expression- Functional differences

- E.g. neural activity in frontal areas tends to be less immediately stimulus-driven and more persistent than primary sensory areas

Paper 4: Neurodynamics of mental exploration

Goal: Better understand how a brain carries out mental exploration

Steps: Design a simple system, based loosely on the rodent hippocampus, capable of mental exploration of possible actions (spatial paths) and choosing a desirable pathway

What is mental exploration?

Mental exploration- Timescale of a fraction of a second to minutes- Involves protracted evolution of neural activity followed

by apt behavioral action directly relating to the activity during the exploration

- Often faster, safer, and more energy-efficient than physical exploration

- Construct mazes α, β, γ, … , into which an animal can be inserted at any location

- Each maze has walls and a floor with a variety of colors, patterns, textures, etc.

- Only local sensory information is available- Only the ensemble of features is a unique descriptor of

place- Over the course of several sessions of experience in each

environment, an animal will become familiar with each environment through passive learning

Model

Experiment- Place a thirsty animal at position w in α, where the

experimenter has now placed a dish of water- Let the animal drink briefly, then remove it from w and

insert it at x, which may or may not be in α- Animal knows water is available at w, but has been placed

elsewhere; no direct way of knowing whether x is connected to w, and water is available, or whether x is in a different maze, and water is not available

Options1. Physical exploration2. Do nothing3. Mental exploration

Neural circuit organization- Arrows are synaptic connection pathways - Areas A and E have excitatory place cells

- Analogous to hippocampal place cells- Respond selectively to spatial location

Results- Learning Multiple Planar “Bump” Attractors- Adaptation Produces Useful Attractor Exploration- Learning an Activity Trajectory and Repeating It Mentally- Motor Controller- Producing a Physical Motion Following a Mental Trajectory- Using Mental Exploration to Choose and Physically Follow

a Path