Learning with spikes, and the Unresolved Question in

Neuroscience/Complex Systems

Tony Bell

Helen Wills Neuroscience Institute

University of California at Berkeley

Learning in real neurons:Long-term potentiation and depression (LTP/LTD)

Bliss & Lomo 1973 discovered associative and input specific (Hebbian) changes in sizes of EPSC’s: a potential memory mechanism (the memory trace). Found first in hippocampus: known to be implicated in learning and memory. LTP from high-frequency presynaptic stimulation, or low-frequency presynaptic stimulation and postsynapticdepolarisation. LTD from prolonged low-frequency stimulation.Levy & Steward (1983) played with timing of weak and stronginput from entorhinal cortex to hippocampus, finding LTD whenweak after strong, LTP when strong up to 20ms after weak or simultaneous.

Spike Timing-Dependent Plasticity (STDP) Markram et al (1997) find 10ms window for time-dependence of plasticity, by manipulating pre- and post-synaptic timings.

Spike Timing Dependent Plasticity

Experimenting with pre- and post-synapticspike-timings at a synapse between a retinal ganglion cell and a tectal cell.(Zhang et al, 1998)

STDP is different in different neurons.

Diverse mechanisms -Common objective??

Figure from Abbott and Nelson

STDP is different in different neurons.

Diverse mechanisms -Common objective??

This may be true,but first we had betterunderstand the mechanism,or we will most likely think up a bad theorybased on our current prejudices and it won’t have any relevance to biology (which, like the rest of the world, is stranger than we can suppose….)

Equation for membrane voltage (cable equation)

… membrane capacitance… conductance along dendrite… maximum conductance for channel species k… time-varying fraction of those channels open… reversal potential for channel species k

Equation for ion channel kinetics (non-linear Markov model)

………. voltage: information from within the cell

………. extracellular ligand: information from other cells

…intracellular calcium: information from other molecules

etc

Can we connect the information-theoretic learning principles we studied yesterday to the biophysical and molecular reality of these processes?

Let’s give it a go in a simplified model….the Spike Response Model (a sophisticatedvariant of the ‘integrate-and-fire’ model).

Gerstner’s SPIKE RESPONSE MODEL:

∑u = k W i j R (t - t )kl k l

l

IMPLICIT DIFFERENTIATION

T = = kl

∂

∂

t

tk

l

W R (t - t )i j kl k l

.

u k.

HOW DOES ONE SPIKE TIMING AFFECT ANOTHER?

The Idea is: output spikes to be as sensitive as possible to inputs. Assuming: a deterministic feedforward invertible network,

Maximum Likelihood: try tomap inputs uniformly into unit hypercube:

p(y) =p(x)

| |

x

y

W

p(y) 1

Maximum Spikelihood:map inputs into independent Poisson processes:

p(t' i' ) =p(t i)

| |

try to

p(t' i')

t iW

t' i'

SPIKELIHOOD

t i

W

t' i'

LIKELIHOOD

x

y

W L(x) = log |W| + ∑ log q(u )i

i

L(t i) > log |T| + ∑ log q(n' )i

~ i

use all firing rates equally

USE THE BANDWIDTH

BE NON-LOSSY

make the spikecount Poisson

OBJECTIVE FUNCTIONS FOR RATE AND SPIKING MODELS:

L(t i) > log |T| + ∑ log q(n' )i

~ i

THE LEARNING RULE

for the objective:

is

mean raterate at input synapse

sum over spikes from neuron j

when T is a single

Simulation results: Coincidence detection (Demultiplexing).

unmixing matrix(learned)

mixing matrix

original spike trains

multiplexed spike trains

demulti-plexed spike trains

time (ms)

A 9x9 network extracts independent point processes from correlated ones

original

demulti-plexed

Identity Unmixing Mixing

× =

Compare with STDP:

Froemke& Dan, Nature 2002

Bell & Parra (NIPS 17)

But real STDP has a predictive component: (spikes also talk about future spikes)

OUT causal

predictive

IN

The Spike Response Model is causal. It only takes into account how output spikes talk about past input spikes:

Postsynaptic calcium integrates this information (Zucker 98), both causal (NMDA channels -> CAM-K) and predictive (L-channels -> calcineurin)?

-requires a non-lossy map [t, i] -> [t, i] (which we enforced…)

-learning is (horrendously) non-local

-model does not match STDP curves

-model ignores predictive information

-information only flows from synapse to soma, and not back down

Problems with this spikelihood model:

in out

By infomaxing from input spikes to output spikes, we are ignoring the information that flows from output spikes (and elsewhere in the dendrites) back down to where the input information came from - the site of learning: the protein/calcium machinery at post-synaptic densities, where the plasticity calculation actually takes place.

What happens if you include this in your Jacobian?

Then the Jacobian between all spike-timings becomes the sum total of all intradendritic causalities. And spikes are talking to synapses, not other spikes. This is a massively overcomplete inter-level information flow (1000 times as many synaptic events as neural events). What kind of density estimation scheme do we then have?

ie: inside the cells:(timings voltage calcium)

models and creates between the cell:(spikes)

The Within models and creates the Between:

neurotransmitter(glutamate)

NMDAchannel

voltage-dependentL-channel

Ca2+ Ca2+

AMPAchannel

synapse

endoplasmicreticulum

dendrite

protein machinery

vesicle with glu receptorsis trafficked to plasma membrane

Post-synaptic machinery (site of learning) integrates incoming spike information with global cell state.

Ca converts timing and voltage information into molecular change2+

network of 2 agents

network of neurons

network of macromolecules

network of protein complexes(eg: synapses)

Networks within networks:

1 cell1 brain

( = STDP)

A Multi-Level View of Learning

LEARNING at a LEVEL is CHANGE IN INTERACTIONS between its UNITS,implemented by INTERACTIONS at the LEVEL beneath, and by extensionresulting in CHANGE IN LEARNING at the LEVEL above.

IncreasingTimescale

Separation of timescales allows INTERACTIONS at one LEVEL to be LEARNING at the LEVEL above.

Interactions=fastLearning=slow

LEVEL UNIT INTERACTIONS LEARNING

society organism behaviour

ecology society predation, symbiosis

natural selection

sensory-motorlearning

organism cell spikes synaptic plasticity

cell

protein molecular forces gene expression,protein recycling

voltage, Ca bulk molecular changessynapse

amino acid

synapse protein direct,V,Ca molecular changes

Advantages:

A closed system can model itself (sleep, thought…)

World modeling is not done directly. Rather, it occurs as a side-effect of self-modeling. The world is a ‘boundary-condition’on this modeling, imposed by the level above - by the social level.

The variables which form the probability model are explicitlylocated at the level beneath the level being modeled.

Generalising to molecular and social networks suggests thatgene expression and reward-based social agency may just be otherforms of inter-level density estimation.

Does the ‘standard model’ really suffice?

ReinforcementDecision

V1

Retina

Thalamus

V whatever

Action Eh..somewhere else

Does the ‘standard model’ really suffice?

ReinforcementDecision

V1

Retina

Thalamus

V whatever

Action Eh..somewhere else

Or is it ‘levels-chauvinism’?

The emerging computational theory of perception is Bayesian inference. It postulates that the sensory system combines a prior probability over possible states of the world, with a likelihood that observed sensory data was caused by each possible state, and computes a posterior probability over the states of the world given the sensory data.

The emerging computational theory of movement is stochastic optimal control. It postulates that the motor system combines a utility function quantifying the goodness of each possible outcome, with a dynamics model of how outcomes are caused by control sequences, and computes a control law (state-control mapping) which optimizes expected utility.

The standard (or rather the slightly more emerged)neurostatistical model, as articulated by Emo Todorov:

But we haven’t seen yet what unsupervised models maydo when they are involved in sensory-motor loops. Theymay sidestep common criticisms of feedforwardunsupervised theories ……

Infomax between Layers.(eg: V1 density-estimates Retina)

Infomax between Levels.(eg: synapses density-estimate spikes)

1 2

• within-level• feedforward• molecular sublevel is ‘implementation’• social superlevel is ‘reward’• predicts independent activity• only models outside input

• between-level• includes all feedback• molecular net models/creates• social net is boundary condition• permits arbitrary activity dependencies• models input and intrinsic together

retina

V1

synaptic weights

x

y all neural spikes

all synaptic readout

synapses,dendites

t

y

pdf of all spike timespdf of all synaptic ‘readouts’

If we canmake thispdf uniform

then we have a model constructed from all synaptic and dendritic causality

This SHIFT in looking at the problemalters the question so that if it isanswered, we have an unsupervised theory of ‘whole brain learning’.

What about the mathematics?Is it tractable?

Not yet.

A new, in many ways satisfactory, objective is defined, but the gradient calculation seems very difficult.

But this is still progress.

by gradient descent in a parameter of the model :

Density Estimation when the input is affected:

Make the model like the reality

by minimising the Kullback-Leibler Divergence:

changing one’s model to fit the world

It is easier to live in a world where one can

change the worldto fit the model, as well as

Conclusion:

This should be easier, but it isn’t yet.

I’m open to suggestions…

What have we learned from other complexself-organising systems?

Is there a simpler model which capturesthe essence of the problem?