Spike-based computation

Slides:



Advertisements
Similar presentations
Introduction to Neural Networks
Advertisements

What is the neural code? Puchalla et al., What is the neural code? Encoding: how does a stimulus cause the pattern of responses? what are the responses.
Spike Train Statistics Sabri IPM. Review of spike train  Extracting information from spike trains  Noisy environment:  in vitro  in vivo  measurement.
Synchrony in Neural Systems: a very brief, biased, basic view Tim Lewis UC Davis NIMBIOS Workshop on Synchrony April 11, 2011.
Neurophysics Part 1: Neural encoding and decoding (Ch 1-4) Stimulus to response (1-2) Response to stimulus, information in spikes (3-4) Part 2: Neurons.
Spike timing-dependent plasticity: Rules and use of synaptic adaptation Rudy Guyonneau Rufin van Rullen and Simon J. Thorpe Rétroaction lors de l‘ Intégration.
1 Testing the Efficiency of Sensory Coding with Optimal Stimulus Ensembles C. K. Machens, T. Gollisch, O. Kolesnikova, and A.V.M. Herz Presented by Tomoki.
Artificial Spiking Neural Networks
Impact of Correlated inputs on Spiking Neural Models Baktash Babadi Baktash Babadi School of Cognitive Sciences PM, Tehran, Iran PM, Tehran, Iran.
What is the language of single cells? What are the elementary symbols of the code? Most typically, we think about the response as a firing rate, r(t),
Functional Link Network. Support Vector Machines.
1 3. Spiking neurons and response variability Lecture Notes on Brain and Computation Byoung-Tak Zhang Biointelligence Laboratory School of Computer Science.
Part II: Population Models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 6-9 Laboratory of Computational.
Romain Brette Computational neuroscience of sensory systems Dynamics of neural excitability.
Neural Coding 4: information breakdown. Multi-dimensional codes can be split in different components Information that the dimension of the code will convey.
Marseille, Jan 2010 Alfonso Renart (Rutgers) Jaime de la Rocha (NYU, Rutgers) Peter Bartho (Rutgers) Liad Hollender (Rutgers) Néstor Parga (UA Madrid)
Introduction to Mathematical Methods in Neurobiology: Dynamical Systems Oren Shriki 2009 Modeling Conductance-Based Networks by Rate Models 1.
1Neural Networks B 2009 Neural Networks B Lecture 1 Wolfgang Maass
Reading population codes: a neural implementation of ideal observers Sophie Deneve, Peter Latham, and Alexandre Pouget.
For stimulus s, have estimated s est Bias: Cramer-Rao bound: Mean square error: Variance: Fisher information How good is our estimate? (ML is unbiased:
Reinagel lectures 2006 Take home message about LGN 1. Lateral geniculate nucleus transmits information from retina to cortex 2. It is not known what computation.
Stable Propagation of Synchronous Spiking in Cortical Neural Networks Markus Diesmann, Marc-Oliver Gewaltig, Ad Aertsen Nature 402: Flavio Frohlich.
Spike-triggering stimulus features stimulus X(t) multidimensional decision function spike output Y(t) x1x1 x2x2 x3x3 f1f1 f2f2 f3f3 Functional models of.
The Decisive Commanding Neural Network In the Parietal Cortex By Hsiu-Ming Chang ( 張修明 )
A globally asymptotically stable plasticity rule for firing rate homeostasis Prashant Joshi & Jochen Triesch
How facilitation influences an attractor model of decision making Larissa Albantakis.
Laurent Itti: CS599 – Computational Architectures in Biological Vision, USC Lecture 7: Coding and Representation 1 Computational Architectures in.
Connected Populations: oscillations, competition and spatial continuum (field equations) Lecture 12 Course: Neural Networks and Biological Modeling Wulfram.
Computing with spikes Romain Brette Ecole Normale Supérieure.
EE513 Audio Signals and Systems Statistical Pattern Classification Kevin D. Donohue Electrical and Computer Engineering University of Kentucky.
Romain Brette Ecole Normale Supérieure, Paris Philosophy of the spike.
Mechanisms for phase shifting in cortical networks and their role in communication through coherence Paul H.Tiesinga and Terrence J. Sejnowski.
Ising Models for Neural Data John Hertz, Niels Bohr Institute and Nordita work done with Yasser Roudi (Nordita) and Joanna Tyrcha (SU) Math Bio Seminar,
Bump attractors and the homogeneity assumption Kevin Rio NEUR April 2011.
Romain Brette Ecole Normale Supérieure, Paris Computing with neural synchrony: an ecological approach to neural computation.
1 / 41 Inference and Computation with Population Codes 13 November 2012 Inference and Computation with Population Codes Alexandre Pouget, Peter Dayan,
The search for organizing principles of brain function Needed at multiple levels: synapse => cell => brain area (cortical maps) => hierarchy of areas.
Lecture 10: Mean Field theory with fluctuations and correlations Reference: A Lerchner et al, Response Variability in Balanced Cortical Networks, q-bio.NC/ ,
Biological Modeling of Neural Networks Week 8 – Noisy input models: Barrage of spike arrivals Wulfram Gerstner EPFL, Lausanne, Switzerland 8.1 Variation.
Cognition, Brain and Consciousness: An Introduction to Cognitive Neuroscience Edited by Bernard J. Baars and Nicole M. Gage 2007 Academic Press Chapter.
2 2  Background  Vision in Human Brain  Efficient Coding Theory  Motivation  Natural Pictures  Methodology  Statistical Characteristics  Models.
Neural coding (1) LECTURE 8. I.Introduction − Topographic Maps in Cortex − Synesthesia − Firing rates and tuning curves.
Romain Brette An ecological approach to neural computation.
Dana Ballard - University of Rochester1 Distributed Synchrony: a model for cortical communication Madhur Ambastha Jonathan Shaw Zuohua Zhang Dana H. Ballard.
$ recognition & localization of predators & prey $ feature analyzers in the brain $ from recognition to response $ summary PART 2: SENSORY WORLDS #10:
The Function of Synchrony Marieke Rohde Reading Group DyStURB (Dynamical Structures to Understand Real Brains)
What is the neural code?. Alan Litke, UCSD What is the neural code?
Neural Modeling - Fall NEURAL TRANSFORMATION Strategy to discover the Brain Functionality Biomedical engineering Group School of Electrical Engineering.
DO LOCAL MODIFICATION RULES ALLOW EFFICIENT LEARNING ABOUT DISTRIBUTED REPRESENTATIONS ? A. R. Gardner-Medwin THE PRINCIPLE OF LOCAL COMPUTABILITY Neural.
6. Population Codes Presented by Rhee, Je-Keun © 2008, SNU Biointelligence Lab,
Ch 9. Rhythms and Synchrony 9.7 Adaptive Cooperative Systems, Martin Beckerman, Summarized by M.-O. Heo Biointelligence Laboratory, Seoul National.
Network Models (2) LECTURE 7. I.Introduction − Basic concepts of neural networks II.Realistic neural networks − Homogeneous excitatory and inhibitory.
Biological Modeling of Neural Networks: Week 10 – Neuronal Populations Wulfram Gerstner EPFL, Lausanne, Switzerland 10.1 Cortical Populations - columns.
Date of download: 6/28/2016 Copyright © 2016 American Medical Association. All rights reserved. From: Teamwork Matters: Coordinated Neuronal Activity in.
The Neural Code Baktash Babadi SCS, IPM Fall 2004.
Biological Modeling of Neural Networks Week 11 – Variability and Noise: Autocorrelation Wulfram Gerstner EPFL, Lausanne, Switzerland 11.1 Variation of.
Biointelligence Laboratory, Seoul National University
The Brain as an Efficient and Robust Adaptive Learner
OCNC Statistical Approach to Neural Learning and Population Coding ---- Introduction to Mathematical.
Synchrony & Perception
Brendan K. Murphy, Kenneth D. Miller  Neuron 
Volume 40, Issue 6, Pages (December 2003)
Volume 36, Issue 5, Pages (December 2002)
Yann Zerlaut, Alain Destexhe  Neuron 
Information Processing by Neuronal Populations Chapter 5 Measuring distributed properties of neural representations beyond the decoding of local variables:
Patrick Kaifosh, Attila Losonczy  Neuron 
The Brain as an Efficient and Robust Adaptive Learner
Volume 74, Issue 1, Pages (April 2012)
Rapid Neocortical Dynamics: Cellular and Network Mechanisms
Patrick Kaifosh, Attila Losonczy  Neuron 
Presentation transcript:

Spike-based computation CA6 – Theoretical Neuroscience romain.brette@ens.fr

Spikes vs. rates

The question Is neural computation based on spikes or on firing rates? Goal of this part: to understand the question!

Three common misconceptions “Both rate and spike timing are important for coding, so the truth is in between” “Neural responses are variable in vivo, therefore neural codes can only be based on rates” “A stochastic spike-based theory is nothing else than a rate-based theory, only at a finer timescale”

Misconception #1: “Both rate and spike timing are important for coding, so the truth is in between”

“Both rate and spike timing are important for coding, so the truth is in between” The « golden mean »: between two extreme positions, an intermediate one must be true. a.k.a. « the golden mean fallacy » Aristotle Extreme Position A: there is a God Extreme Position B: there is no God => there is half a God! Are rate-based and spike-based views two extreme positions of the same nature?

Of spikes and rates Spikes: a well-defined timed event, the basis of neural interaction Rates: an abstract concept defined on spikes dt e.g. temporal or spatial average (defined in a large N limit); probabilistic expectation. Rate-based postulate: this concept/approximation captures everything relevant about neural activity Spike-based view: this postulate is not correct This does not mean that « rate » is irrelevant!

Rate in spike-based theories Spike-based computation requires spikes More spikes, more computation Therefore, firing rate determines quantity of information Spike-based view: rate determines quantity of information Rate-based view: rate determines content of information

The tuning curve Firing rate varies with stimulus properties (rate-based) Firing rate « encodes » direction or: (spike-based) The neuron spends more energy at the « preferred » direction (rate is a correlate of computation) The question is not: « is firing rate or spike timing more informative/useful? » but: « which one is the basis of computation? »

“Both rate and spike timing are important for coding, so the truth is in between” Spike-based view: rate determines quantity of information Rate-based view: rate determines content of information

Misconception #2: “Neural responses are variable in vivo, therefore neural codes can only be based on rates”

Neural variability Temporal irregularity Close to Poisson statistics Softky & Koch, J Neuroscience (1993) ISI rate (Hz), V1 Close to Poisson statistics Rate-based view: spike trains have Poisson statistics (ad hoc hypothesis) Spike-based view: spike trains have Poisson statistics (maximum information) Lack of reproducibility - empirically questionable - could result from uncontrolled variable But let’s assume it’s true and examine the argument!

No reproducibility => rate-based? lack of reproducibility => either stochastic or chaotic This is about stochastic/chaotic vs. deterministic, not about rate-based vs. spike-based Implicit logic responses of N neurons are irreproducible => there exist N dynamic quantities that completely characterize the state of the system and its evolution determine the probability of firing of the neurons This is pure speculation!

A counter-example Sparse coding Imagine you want to code this signal: with the spike trains of N neurons, so that you can reconstruct the signal by summing the PSPs 𝑆 𝑡 ≈ 𝑖,𝑗 𝑃𝑆𝑃(𝑡− 𝑡 𝑖 𝑗 ) (with a given rate) The problem is degenerate, so there are many solutions. For example this one: Or this one:

A counter-example 𝑆 𝑡 ≈ 𝑖,𝑗 𝑃𝑆𝑃(𝑡− 𝑡 𝑖 𝑗 ) 𝑆 𝑡 ≈ 𝑖,𝑗 𝑃𝑆𝑃(𝑡− 𝑡 𝑖 𝑗 ) The problem is degenerate, so there are many solutions. For example this one: Or this one: It is variable It cannot be reduced to rates, because error is in 1/N, not 1/√N

The argument strikes back Do rate-based theories account for neural variability? Rate-based theories are deterministic Deterministic description is obtained by averaging, a.k.a. removing variability Rate-based theories do not account for neural variability, they acknowledge that there is neural variability To account for variability of spike trains requires spikes, i.e., a stochastic/chaotic spike-based theory

“Neural responses are variable in vivo, therefore neural codes can only be based on rates” Rate-based theories do not account for neural variability, they acknowledge that there is neural variability, and postulate that it is irrelevant (averaging) To account for variability of spike trains requires spikes, i.e., a stochastic/chaotic spike-based theory

Misconception #3: “A stochastic spike-based theory is nothing else than a rate-based theory, only at a finer timescale”

“A stochastic spike-based theory is nothing else than a rate-based theory, only at a finer timescale” spikes In terms of stimulus-response properties, there is about the same information in the time-varying rate rate Rate-based postulate: for each neuron, there exists a private quantity r(t) whose evolution only depends on the other quantities ri(t). spike trains are derived from r(t) only r1 stochastic r2 r = f(r1, r2, rn) It is assumed that this is approximately the same for all realizations rn

“A stochastic spike-based theory is nothing else than a rate-based theory, only at a finer timescale” Rate-based postulate: for each neuron, there exists a private quantity r(t) whose evolution only depends on the other quantities ri(t). spike trains are derived from r(t) only r1 stochastic r2 r = f(r1, r2, rn) It is assumed that this is approximately the same for all realizations rn Implication: spike trains are realizations of independent random processes, with a source of stochasticity entirely intrinsic to the neuron. This has nothing to with the timescale!

Reformulating the question

Is neural computation based on spikes or on firing rates? Can neural activity and computation be entirely and consistently described by the dynamics of time-varying rates in the network? Spelling out the rate-based postulate for each neuron, there exists a private quantity r(t) whose evolution only depends on the other quantities ri(t). ri(t) is the expected firing probability of neuron i. spike trains (realizations) depend on r(t) only, through a private stochastic process (independent neurons)

Spelling out the rate-based postulate for each neuron, there exists a private quantity r(t) whose evolution only depends on the other quantities ri(t). ri(t) is the expected firing probability of neuron i. spike trains (realizations) depend on r(t) only, through a private stochastic process (independent neurons) Example 1: random networks If true, then ri(t) can be found by writing self-consistent equations (cf. work by Nicolas Brunel and coll.) This works for sparse random networks, but not in general. Example 2: sparse coding 𝑆 𝑡 ≈ 𝑖,𝑗 𝑃𝑆𝑃(𝑡− 𝑡 𝑖 𝑗 ) Signal reconstruction is more accurate than with rates

Conclusion The rate-based postulate for each neuron, there exists a private quantity r(t) whose evolution only depends on the other quantities ri(t). ri(t) is the expected firing probability of neuron i. spike trains (realizations) depend on r(t) only, through a private stochastic process (independent neurons) This is mainly a methodological postulate (=convenient). Does not derive from observations of neural variability, or stochasticity

Flavors of spike-based computation Theories based on synchrony Synfire chains (Abeles) Polychronization (Izhikevich) Synchrony as sensory invariant (Brette) Theories based on asynchrony Rank order coding (Thorpe) Sparse coding (Olshausen, Lewicki) Predictive coding (Denève)

Spike-based computation (I): Synchrony

Motivation: what makes a neuron spike? The coding metaphor Neural representations in the mind of the observer. But the neuron is not an arbitrary observer!

Motivation: what makes a neuron spike? The acting metaphor The neuron acts on its environment Neural representations in the mind of the observer. What makes a neuron spike: the rate of its inputs, and/or the relative timing between them?

Integration vs. coincidence detection Viewpoint #1: neurons respond to coincident spikes threshold threshold spike no spike but this is with 2 spikes!

Integration vs. coincidence detection Viewpoint #2: neurons respond to the mean input Law of large numbers: A cortical neuron integrates many inputs (about 10,000) If these inputs are independent, their sum should not be very variable (law of large numbers) Mean total input = determined by input firing rates, not by coincidences Which viewpoint is correct?

The fluctuation-driven or « balanced » regime

Irregularity of spike trains Cortical neurons fire irregularly (in vivo) Coefficient de variation Softky & Koch (1993) [cortex visuel]

The paradox of irregular spike trains A cortical neuron integrates many inputs (about 10,000) If these inputs are independent, their sum should not be very variable (law of large numbers) Therefore firing should be regular! total input threshold

Campbell theorem Consider a Poisson process {ti} with rate F. Consider a linear superposition of PSPs: Then: « shot noise » This works with other processes Comme le calcul avec les mains qu’on a fait avant. This is only true for Poisson processes (same formulae applies to postsynaptic currents/conductances)

Mean input and threshold Mean-driven regime Small variability (average of many inputs) Regular firing (CV close to 0)

Mean input and threshold Fluctuation-driven regime (« balanced regime ») spikes can only occur at times when the input fluctuates above the mean -> irregular firing (CV close to 1)

Membrane potential distribution in vivo Rossant et al. (2011) J Neurosci Membrane potential distribution peaks below threshold CV is high -> fluctuation-driven regime is typical

Why « balanced regime »?

The balanced regime A few figures for a cortical neuron: 10,000 synapses, 80% excitatory, 20% inhibitory PSP 5-10 mV.ms Average spontaneous firing rate  1 – 4 Hz Mean depolarization (Campbell): 40 – 300 mV Standard deviation: 1.5 – 4.5 mV (assuming independency) Difference between rest and threshold: 10-20 mV 0.5 mV (Mason, Nicoll & Stratford, J Neurosci 2001; cortex du rat in vitro) 10-20 ms excitation Why it is called the balanced regime (in vivo, the time constant is smaller, giving a smaller integral); cortex somatosensoriel du rat See also Shadlen & Newsome, Curr Op in Neurobiology (1994) Ecart-type d’après le théorème de Campbell (shot noise) Compensation excitation inhibition: confirmé par des mesures To get a subthreshold mean depolarisation, inhibition must balance excitation: « balanced regime » or « fluctuation-driven »

Coincidence detection in The fluctuation-driven regime

Elementary remark Fluctuation-driven regime (« balanced regime ») Output firing rate is not determined by the mean input, otherwise Fout=0 Hz. Therefore relative spike timing (variance) does matter. But how much?

An example (IF model) 4000 independent Poisson excitatory inputs + 1000 inhibitory inputs (balanced) Pairwise correlation: P(j spikes | i spikes at the same time) = P(i is in a synchrony event) * P(j is in the same synchrony event) = (400 synchrony spikes / 4000 total spikes)*(9/3999) Not experimentally detectable, and therefore consistent with experimental findings! synchrony events involving 10 random synapses, at rate 40 Hz Pairwise correlation: 0.0002 (probably not experimentally detectable!)

Why tiny correlations may have large postsynaptic effects In a balanced regime, output rate depends on both mean and variance of the input Consider N random variables Xn with identical distributions and correlation c. What is the variance of S? if c =0 otherwise correlations can be neglected only if c << 1/N.

Asynchronous spikes vs. coincident spikes 2 input spikes in a noisy neuron 2 coincident spikes Coincidence sensitivity S = difference (0.11) Rossant et al. (2011) Sensitivity of Noisy Neurons to Coincident Inputs. J Neuroscience

Fluctuation-driven regime 2 spikes p spikes distribution of Vm Turns out that it works very well, because the underestimation we make in the two quantities are similar, and therefore disappears with the subtraction. p simultaneous spikes are much more efficient than p non-coincident spikes!

Mean-driven regime 2 coincident spikes are (slightly) less efficient than 2 non-coincident spikes!

Neurons are sensitive to coincidences in the balanced regime only (= fluctuation-driven) Oscillator regime (= mean-driven)

Distributed synchrony Synchrony events involving p random synapses (w=0.5 mV) Top: independent and synchrony events Bottom: theory & simulation Cortical neurons in vitro:

Distributed synchrony Synchrony events involving p random synapses (w=0.5 mV) Top: independent and synchrony events Bottom: theory & simulation Compare with changing the firing rate of p inputs

Summary In the fluctuation-driven regime, neurons are extremely sensitive to input correlations Correlations have negligible effects only if they are small compared to 1/N (N synapses) In fact, correlations undetectable in pair recordings can have tremendous postsynaptic effect Rate-based computation would require very specific mechanisms to cancel correlations! Rossant et al. (2011) Sensitivity of Noisy Neurons to Coincident Inputs. J Neuroscience

Synfire chains

Synfire chains Concept developed by Moshe Abeles (1982); similar idea in Griffith (Biophysical Journal 1963) see the book: Corticonics: Neural Circuits of the Cerebral Cortex (1991) feedforward structure (could be embedded in recurrent network) (Diesmann et al, Nature 1999) simultaneous activation of neurons in a layer -> propagation from layer to layer simulation with noisy IF models

Synfire chains: Stability analysis (Diesmann et al, Nature 1999) layer 1 layer 2 synchronous propagation a = number of spikes standard deviation  attractor dissipation Trajectories in (,a) space

Synfire chains: computation Probability of firing = function of weighted sum of binary inputs = discrete-time formal neural network Note: processing time is faster than membrane time constant (here τ =15 ms, transmission delays = 5 ms)

Functional motivation: compositionality and the binding problem blue shape area red square color area disk blue ? shape area red square color area disk

Binding by oscillations See work by Wolf Singer. Gamma oscillations gamma in cortex (50 Hz) Hypothesis: properties of the same object are encoded by spikes within the same period of an oscillation bleu Hypothèse controversée (Singer etc) rouge it’s a blue square! carré rond time as a signature of objects

Binding by oscillations See work by Wolf Singer. Gamma oscillations gamma in cortex (50 Hz) Hypothesis: properties of the same object are encoded by spikes within the same period of an oscillation bleu Hypothèse controversée (Singer etc) rouge it’s a not a blue square! carré rond Only properties encoded in the same oscillation period can interact

General point The concept of « neural assembly » (= set of neurons) has a weak structure = like « bag of words » used in search engines of general concept point a used structure set like neurons weak search has engines assembly bag in The neural

General point The concept of « neural assembly » (= set of neurons) has a weak structure = like « bag of words » used in search engines The binding problem requires something like this: « binding by synchrony » = time plays the role of a label See work by Christof von der Malsburg

General point « binding by synchrony » = time plays the role of a label Note that this is not sufficient to represent complex structures. Example: « cat eats mouse » and « mouse eats cat ». mouse cat who eats who? eats

Polychronization

Polychronization Extension of synfire chains with axonal delays Term introduced by Izhikevich (Izhikevich, Neural Comp 2006; Szatmary and Izhikevich, PLoS CB 2010) Previous work by Bienenstock (1994) under the name « synfire braid » Izhikevich, Neural Computation 2006 simultaneous activation of b, c, d: no propagation propagation to a propagation to e -> the same neurons can be involved in different « polychronous groups », depending on relative time of activation -> many more polychronous groups than synfire chains

Polychronization Toy example: 5 neurons 14 polychronous groups

Polychronization and working memory Szatmary and Izhikevich, PLoS Computational Biology 2010 Neurons are activated in a spatiotemporal pattern congruent to a polychronous group -> connections reinforce (short-term STDP) -> activity is spontaneously replayed

Polychronization and working memory Szatmary and Izhikevich, PLoS Computational Biology 2010 Conceptual context: Edelman’s neural darwinism theory (Neuron 1993) inspired from immune system many polychronous groups (=potential memories) exist a particular activation pattern strengthens (« selects ») one polychronous group through associative plasticity

Synchrony as sensory invariant

The synchrony receptive field Starting point: a neuron fires when (some of) its presynaptic neurons fire simultaneously no response When does this happen? « Synchrony receptive field » : the set of stimuli that elicit synchronous firing in a given set of neurons Note: the SRF should be defined from the postsynaptic point of view, i.e., including axonal delays Brette (2012). Computing with synchrony. PLoS Comp Biol

Example Synchrony (from postsynaptic viewpoint) when: S(t-dR-δR)=S(t-dL-δL) dR-dL = δL - δR Independent of source signal The synchrony receptive field corresponds to a sensory invariant or « law » connects to James Gibson’s « invariant structure » (book « The Ecological Approach to Visual Perception »)

Synchrony receptive field as sensory law B no response « Synchrony receptive field » = {S | NA(S) = NB(S)} = a law followed by sensory signal S(t)

Implicit assumption: reproducibility of spike timing In spiking model: Z. Mainen, T. Sejnowski, Science (1995) Spike timing is reproducible in vitro for time-varying inputs Z. Mainen, T. Sejnowski, Science (1995) Brette, R. and E. Guigon (2003). Reliability of spike timing is a general property of spiking model neurons. Neural Comput Brette (2012). Computing with neural synchrony. PLoS Comp Biol

Shared variability What needs to be reproducible is relative, not absolute, spike timing. shared input (e.g. modulation by attention) that is not stimulus-locked absolute not reproducible but relative timing not affected Brette (2012). Computing with neural synchrony. PLoS Comp Biol

Non-trivial example: binaural hearing in real life FR,FL = location-dependent acoustical filters (HRTFs/HRIRs) Delay: low frequency high frequency Sound propagation is more complex than pure delays!

Non-trivial example: binaural hearing in real life FR,FL = location-dependent acoustical filters (HRTFs/HRIRs) Delay: low frequency high frequency Frequency-dependent Interaural Time Differences: FRONT BACK ITD (ms) Frequency

Binaural sensory laws in real life Binaural stimulus: SR = FR*S SL= FL*S Sensory laws followed by the stimulus: FL*SR = FR*SL or U*FL*SR = U*FR*SL for any filter U

Binaural synchrony receptive fields FR,FL = HRTFs/HRIRs (location-dependent) NA, NB = neural filters (e.g. basilar membrane filtering) input to neuron A: NA*FR*S (convolution) input to neuron B: NB*FL*S Synchrony when: NA*FR = NB*FL SRF(A,B) = set of filter pairs (FL,FR) = set of source locations = spatial receptive field Independent of source signal S

Decoding synchrony structure basilar membrane MSO cochlear nucleus Each source location is represented by a specific assembly of binaural neurons = neurons whose inputs contain the location in their SRF

Proof of concept Sounds: noise, musical instruments, voice (VCV) Gammatone filterbank + more filters Spiking: noisy IF models Coincidence detection: noisy IF models Acoustical filtering: measured human HRTFs Activation of all assemblies as a function of preferred location: Goodman DF and R Brette (2010). Spike-timing-based computation in sound localization. PLoS Comp Biol 6(11): e1000993. doi:10.1371/journal.pcbi.1000993.

The biological hypothesis Each binaural neuron encodes an element of binaural structure NB*FL*S NA*FR*S Each binaural neuron encodes a particular element of binaural structure, for a specific frequency band and location FL*S FR*S

Spike-based computation (II): Asynchrony

General motivation: asynchrony in neural networks Empirically: pairwise correlations between neurons are often small (Ecker et al., « Decorrelated neuronal firing in cortical microcircuits », Science 2010) Conceptually: synchronous firing is redundant, and therefore less efficient in terms of « coding »

Is it contradictory with synchrony-based ideas? No! In synchrony-based schemes: synchrony is a meaningful event, therefore rare synfire chains: groups of neurons that fire together change polychronization: neurons are not synchronous; polychrony is transient synchrony as sensory-invariant: synchrony is stimulus-specific and group-specific, not widespread Distinguishing feature: in synchrony-based schemes, coincidence detection is a core operation

Rank order coding

Neural « codes » Code: spike count (rate code) decoding→integration Timing Rank Code: spike count (rate code) decoding→integration spike timing (temporal code) decoding→coincidence detection spike order (rank order code) decoding→? (Thorpe et al 2001) Comment les neurones représentent-ils les stimuli en trains de potentiels d’action? NB: tps représenté à l’envers ici (1er spike à droite, en fait c’est les spikes voyageant le long des axones qui sont représentés) Thorpe et al. “Spike-based strategies for rapid processing”, Neural Networks (2001)

Decoding rank order How to distinguish between AB and BA? Solution: excitation and inhibition A B Excitatory PSP Inhibitory PSP - - + +

Prey localization by the sand scorpion Inhibition of opposite neuron → more spikes near the source (polar representation of firing rates) Conversion rank order code → rate code Stürtzl et al. (2000). Theory of arachnid prey localization. PRL

Sparse coding

Spikes are very costly P. Lennie. The Cost of Cortical Computation. Current Biology, Vol. 13, 493–497 (2003) Entire body Entire brain Cortex

Sparse coding Encoding: Iterative procedure: add spikes until a target precision is reached i.e., minimize the number of spikes to encode the sound (algorithm= « matching pursuit ») What spikes encode for 3 neurons: Smith & Lewicki (Nature 2006)

Properties of sparse spike codes Reconstruction error scales a 1/N Spike timing matters Spikes are coordinated: if one neuron fails to spike (noise), then other neurons must compensate for it Reproducibility is stimulus-dependent: high if stimulus is complex, low if stimulus is redundant (=degenerate problem)

Lack of reproducibility with deterministic sparse codes Imagine you want to code this signal: with the spike trains of N neurons, so that you can reconstruct the signal by summing the PSPs 𝑆 𝑡 ≈ 𝑖,𝑗 𝑃𝑆𝑃(𝑡− 𝑡 𝑖 𝑗 ) (with a given rate) The problem is degenerate, so there are many solutions. For example this one: Or this one:

Predictive coding

Predictive coding and spike-based inference input output linear read-out Define a read-out of the output neurons Define an error criterion on this read-out Define neuron dynamics so that a spike is produced when it reduces the error Spikes are decisions can be applied to sparse coding (error = reconstruction error + firing rate) References: Boerlin & Denève (2011). Spike-Based Population Coding and Working Memory. PLoS Comp Biol S Denève (2008). Bayesian spiking neurons I: inference. Neural Computation S Denève (2008). Bayesian spiking neurons II: learning. Neural Computation

Summary Theories based on synchrony Synfire chains (Abeles) Polychronization (Izhikevich) Synchrony as sensory invariant (Brette) Theories based on asynchrony Rank order coding (Thorpe) Sparse coding (Olshausen, Lewicki) Predictive coding (Denève)