Modelling stochastic neural learning

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1 Modelling stochastic neural learning Computational Neuroscience András Telcs Compiled from lectures of Prashant Joshi Jotahna Pillow Review of Statistically optimal perception and learning: from behavior to neural representations. Fiser, Berkes, Orban & Lengyel Trends in Cognitive Sciences (2010) Spontaneous Cortical Activity Reveals Hallmarks of an Optimal Internal Model of the Environment. Berkes, Orban, Lengyel, Fiser. Science (2011) Neural Dynamics as Sampling: A Model for Stochastic Computation in Recurrent Networks of Spiking Neurons Lars Buesing, Johannes Bill, Bernhard Nessler, Wolfgang Maass Published: November 3, 2011 DOI: /journal.pcbi

2 Brain - What For? Thoughts Feelings Learning Consciousness Decisions Sensory Inputs Motor Outputs

3 Mammalian Brains

4 Broad Classification of Brain

5 Each Region Has a Specific Function Spinal Cord - Sensory & motor information and control of reflexes Brainstem - Information gateway between body and brain, integrative functions (cardiovascular, respiratory, and pain-sensitivity control and also consciousness) Cerebellum - Motor Control, Attention & Language etc. Diencephalon (Thalamus) - Inputs to cortex Diencephalon (Hypo-Thalamus) - Control of endocrine system (hormones) Prosencephalon (Hippocampus) - Learning, long term consolidation Prosencephalon (Cortex) - Higher cognitive functions

6 Brodmann's Map

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8 Anatomy of a Single Neuron Cell Body (Soma) - Contains DNA, cell nucleus Dendrite - Brings input to the neuron Axon - The output of a neuron Synapse - Transfer the information from one cell to another

9 Ion Channels Physiological Specialization - Membrane spanning ion channels Allow ions to move in and out of cell Work by opening and closing in response to voltage changes Most prominent being Na+, K+, Ca2+, Cl-

10 Spikes - Units of Information Transfer If a neuron is depolarized sufficiently to raise the membrane potential above a threshold level, a spike or action potential is generated Roughly 100 mv fluctuation, lasts for ~ 1ms Of great importance, as spikes can propagate over large distances A spike causes a neuron to go into refractory period

11 Rough Estimates In human brain there are: ~ (trillion) Neurons ~ (quadrillion) Synapses ~ 10 5 Neurons/mm 3 ~ 10 9 Synapse/mm 3 ~ 4 Km Axon/mm 3 ~ 500 million dendrites /mm 3 ~ 10 4 Input Synapses / neuron

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16 Spike train, spike raster

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19 a action/respond

20 Representation learning encoding decoding ere

21 Task learning about the black box s r Stimulus Spike/response r t rs( t,..., t)

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38 Synaptic weights

39 Synaptic weights

40 Synaptic weights

41 Synaptic weights If neuron A frequently contributes to fire neuron B then the synaptic connection /weight strengthen/increased.

42 Synaptic weights If neuron A frequently contributes to fire neuron B then the synaptic connection /weight strengthen/increased. - synaptic plasticity

43 How to store/represent and sample a distribution? p z,..., 1 z k Joint distr. of k binary variables.

44 How to store/represent and sample a distribution? p z,..., 1 z k Joint distr. of k binary variables. We construct a network N of k spiking neurons i i=1,,k which produce sample from p. The firing of N will be a non-reversible MC.

45 Neural representation Spike/action potential s i with 1 ms duration Influence on itself and other neurons last ms refactory period

46 Neural representation Spike/action potential s i with 1 ms duration Influence on itself and other neurons last ms refactory period Time is discretized. Unit is small, an integer time constant is defined multiple of that unit.

47 Neural representation Spike/action potential s i with 1 ms duration Influence on itself and other neurons last ms refactory period Time is discretized. Unit is small, an integer time constant is defined multiple of that unit. z i t 1 s i fired in t ( t, t ] i.e. The rv is set to 1 for duration of, for the refractory period of the neuron.

48 Neural representation The Markovity needs the bookkeeping of the times: We define for each i i denote the time when the neuron spike whitin the interval, ( t, t ],... R k 1 k

49 Neural representation The Markovity needs the bookkeeping of the times: We define for each i i denote the time when the neuron spike whitin the interval, ( t, t] and setting z i = 1 at time t.,... R k 1 k

50 Neural representation The Markovity needs the bookkeeping of the times: We define for each i i denote the time when the neuron spike whitin the interval, ( t, t] and setting z i = 1 at time t.,... R k 1 k i time count down: at the spike, the variable i is set to then decreasing linearly above 0.

51 Neural representation of a pd The membrane potential of the i-th neuron is u i (t) at time t. u i t log P P t t and can be computed by the neuron (neuronal computability condition) Notation z i, z \i z z i i 1 0 z z \ i \ i

52 i i i j i j i j i z b z z w Z z p,, exp 1 w, b parameters w i,j =w j,i, w i,i =0, Examples: Boltzmann distributions

53 Neural representation of a pd u i t b w i i, j j z j t b i individual bias (excitability) w i,j synaptic weight from i to j Random spikes will be triggered by u i and i. p z 1 Z exp z T u t

54 MC in discrete time The time variable i when i spikes is set to (>0 integer). The neuron i can spike iff i <=1. If i >1 the neuron in refractory period and at each time step i is decreased by one.

55 MC in discrete time We define p(,z) so that z p z p, and the Markov transition to ensure that p(z) is the stationary distribution

56 MC in discrete time p, z p zpz k z p z p p p p j 1 k 1 k, z 1 1 j j j j if k>1 0,1, z 0 u log j j j 0 1, z 0 1 u log j j j j j j

57 MC in discrete time p z j j

58 MC in discrete time p p z j z j j j

59 MC in discrete time p p z j z j 0 z j j j j p In all other cases

60 MC in discrete time the deterministic and random transition of,z = (u k -log ) 1-1- Z=1 in all the red circled states Z=0 for the single case left

61 MC in discrete time The dynamic is defined by the map T which is simply T T k T k 1... T 1 that is, the neuron states updated in a fixed order and at the current step, the membrane potential is based on the state of the updated or non-updated union of the set\i furthermore the new state i depends on the previous i prec.

62 MC in discrete time The variable z i is determined by i deterministically z i 1 if 1 i and z i 0 if 0 i

63 Result 1. T i leaves invariant p(,z), 2. Any combination, and T as well does. 3. T defines an aperiodic, irreducible MC has unique invariant distribution 4. p(,z) the unique invariant distribution. 5. p(z) can be sampled via p(,z) using the MC.

64 Extensions 1. relative refractory state 2. continuous time 3. other distributions

65 Extensions 1. relative refractory state the readiness function g is introduced, the sigmoid should be replaced by a proper function f satisfying: exp u f u t1 st1 s1 Remark. The pd is approximated only. 1 1 g g s f u s f u

66 Simulation Demonstration of probabilistic inference, perceptual multistability, binocular rivalry

67 Simulation

68 Simulation

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70 Thanks for the attention! Q?

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