1. Spike-based computation
CA6 – Theoretical Neuroscience

2. Spikes vs. rates

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

4. 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”

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

6. “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?

7. 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!

8. 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

9. 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? »

10. “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

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

12. 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!

13. 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!

14. 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:

15. 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

16. 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

17. “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

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

19. “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

20. “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!

21. Reformulating the question

22. 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)

23. 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

24. 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

25. 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)

26. Spike-based computation (I): Synchrony

27. 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!

28. 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?

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

30. 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?

31. The fluctuation-driven or « balanced » regime

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

33. 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

34. 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)

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

36. 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)

37. 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

38. Why « balanced regime »?

39. 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: 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 »

40. Coincidence detection in The fluctuation-driven regime

41. 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?

42. An example (IF model)
4000 independent Poisson excitatory inputs 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: (probably not experimentally detectable!)

43. 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.

44. 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

45. 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!

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

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

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

49. 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

50. 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

51. Synfire chains

52. 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

53. 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

54. 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)

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

56. 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

57. 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

58. 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

59. 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

60. 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

61. Polychronization

62. 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

63. Polychronization
Toy example: 5 neurons
14 polychronous groups

64. 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

65. 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

66. Synchrony as sensory invariant

67. 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

68. 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 »)

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

70. 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

71. 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

72. 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!

73. 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

74. 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

75. 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

76. 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

77. 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): e doi: /journal.pcbi

78. 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

79. Spike-based computation (II): Asynchrony

80. 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 »

81. 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

82. Rank order coding

83. 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)

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

85. 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

86. Sparse coding

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

88. 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)

89. 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)

90. 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:

91. Predictive coding

92. 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

93. 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)