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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