Biological Modeling of Neural Networks:
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Transcript of Biological Modeling of Neural Networks:
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Biological Modeling of Neural Networks:
Week 14 – Dynamics and Plasticity
Wulfram GerstnerEPFL, Lausanne, Switzerland
14.1 Reservoir computing - Complex brain dynamics - Computing with rich dynamics14.2 Random Networks
- stationary state - chaos
14.3 Stability optimized circuits - application to motor data14.4. Synaptic plasticity - Hebbian - Reward-modulated 14.5. Helping Humans - oscillations - deep brain stimulation
Week 14 – Dynamics and Plasticity
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Neuronal Dynamics – Brain dynamics is complex
motor cortex
frontal cortex
to motoroutput
10 000 neurons3 km wire
1mm
Week 14-part 1: Review: The brain is complex
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Neuronal Dynamics – Brain dynamics is complex
motor cortex
frontal cortex
to motoroutput
Week 14-part 1: Review: The brain is complex
-Complex internal dynamics-Memory-Response to inputs-Decision making-Non-stationary-Movement planning-More than one ‘activity’ value
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Liquid Computing/Reservoir Computing: exploit rich brain dynamics
Readout 1
Readout 2
Maass et al. 2002,Jaeger and Haas, 2004Review:Maass and Buonomano,
Stream of sensory inputs
Week 14-part 1: Reservoir computing
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See Maass et al. 2007
Week 14-part 1: Reservoir computing
-‘calculcate’ 3 4 - if-condition on 1
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ModelingHennequin et al. 2014,See also:Maass et al. 2002,Sussillo and Abbott, 2009Laje and Buonomano, 2012Shenoy et al., 2011
Experiments ofChurchland et al. 2010Churchland et al. 2012See also:Shenoy et al. 2011
Week 14-part 1: Rich dynamics Rich neuronal dynamics
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-Long transients
-Reliable (non-chaotic)
-Rich dynamics (non-trivial)
-Compatible with neural data (excitation/inhibition)
-Plausible plasticity rules
Week 14-part 1: Rich neuronal dynamics: a wish list
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Biological Modeling of Neural Networks:
Week 14 – Dynamics and Plasticity
Wulfram GerstnerEPFL, Lausanne, Switzerland
14.1 Reservoir computing - Complex brain dynamics - Computing with rich dynamics14.2 Random Networks
- rate model - stationary state and chaos
14.3 Hebbian Plasticity - excitatory synapses - inhibitory synapses14.4. Reward-modulated plasticity - free solution 14.5. Helping Humans - oscillations - deep brain stimulation
Week 14 – Dynamics and Plasticity
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I(t)
)(tAn
Week 14-part 2: Review: microscopic vs. macroscopic
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Homogeneous network:-each neuron receives input from k neurons in network-each neuron receives the same (mean) external input
Week 14-part 2: Review: Random coupling
excitation
inhibition
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Stochastic spike arrival: excitation, total rate Re
inhibition, total rate Ri
u0u
EPSC IPSC
Synaptic current pulses
)()()( ttIRuuudt
d meaneq
Langevin equation,Ornstein Uhlenbeck process Fokker-Planck equation
)()()(','
''
, fk
fki
fk
fkeeq ttqttqRuuu
dt
d
Week 14-part 2: Review: integrate-and-fire/stochastic spike arrival
Firing times:Threshold crossing
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( )i i ij jj
dr r F w r
dt
Fixed point with F(0)=0 0ir
stable unstable
Suppose:1 '(0) ( 0)
dF F x
dx
( )dx x F w x
dt
Suppose 1 dimension
Week 14-part 2: Dynamics in Rate Networks F-I curve of rate neuron
Slope 1
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Exercise 1: Stability of fixed point
Next lecture: 9h43( )
dx x F w x
dt
Fixed point with F(0)=0 0x Suppose:
1 '(0) ( 0)d
F F xdx
( )dx x F w x
dt
Suppose 1 dimension
Calculate stability, take w as parameter
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( )i i ij jj
dr r F w r
dt
Fixed point with F(0)=0 0ir
stable unstable
Suppose:1 '(0) ( 0)
dF F x
dx
( )dx x F w x
dt
w<1 w>1
Suppose 1 dimension
Week 14-part 2: Dynamics in Rate Networks
Blackboard:Two dimensions!
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( )i i ij jj
dr r F w r
dt
Fixed point:0ir
stable unstable
Chaotic dynamics: Sompolinksy et al. 1988 (and many others:Amari, ...
1 '(0) ( 0)d
F F xdx
Random,10 percent connectivity
Re( ) 1 Re( ) 1
Week 14-part 2: Dynamics in RANDOM Rate Networks
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Rajan and Abbott, 2006Image: Ostojic, Nat.Neurosci, 2014
( ) ( )i i ij j ij
dr r F w r t
dt
Unstable dynamics and Chaos
Image: Hennequin et al. Neuron, 2014
chaos
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Image: Ostojic, Nat.Neurosci, 2014
( )fi i ij jj f
du u w t t
dt
Firing times:Threshold crossing
Week 14-part 2: Dynamics in Random SPIKING Networks
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Ostojic, Nat.Neurosci, 2014
Week 14-part 2: Stationary activity: two different regimes
Switching/bursts long autocorrelations:Rate chaos
Re( ) 1 Stable rate fixed point,microscopic chaos
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-Long transients
-Reliable (non-chaotic)
-Rich dynamics (non-trivial)
-Compatible with neural data (excitation/inhibition)
-Plausible plasticity rules
Week 14-part 2: Rich neuronal dynamics: a wish list
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Biological Modeling of Neural Networks:
Week 14 – Dynamics and Plasticity
Wulfram GerstnerEPFL, Lausanne, Switzerland
14.1 Reservoir computing - Complex brain dynamics - Computing with rich dynamics14.2 Random Networks
- stationary state - chaos
14.3 Stability optimized circuits - application to motor data14.4. Synaptic plasticity - Hebbian - Reward-modulated 14.5. Helping Humans - oscillations - deep brain stimulation
Week 14 – Dynamics and Plasticity
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Image: Hennequin et al. Neuron, 2014
Re( ) 1 Re( ) 1
Week 14-part 3: Plasticity-optimized circuit
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Optimal control of transient dynamics in balanced networkssupports generation of complex movements
Hennequin et al. 2014,
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Random stability-optimized circuit (SOC)
Random connectivity
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Random
Week 14-part 3: Random Plasticity-optimized circuit
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studydurationof transients
F-I curve of rate neuron
slope 1/linear theory
slope 1/linear theory
a1= slowest = most amplified
Week 14-part 3: Random Plasticity-optimized circuit
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slope 1/linear theoryF-I curve of rate neuron
a1= slowest = most amplified
Week 14-part 3: Random Plasticity-optimized circuit
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Optimal control of transient dynamics in balanced networkssupports generation of complex movements
Hennequin et al. NEURON 2014,
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Churchland et al. 2010/2012 Hennequin et al. 2014
Week 14-part 3: Application to motor cortex: data and model
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Quiz: experiments of Churchland et al.
[ ] Before the monkey moves his arm, neurons in motor-related areas exhibit activity[ ] While the monkey moves his arm, different neurons in motor- related area show the same activity patterns [ ] while the monkey moves his arm, he receives information which of the N targets he has to choose [ ] The temporal spike pattern of a given neuron is nearly the same, between one trial and the next (time scale of a few milliseconds)[ ] The rate activity pattern of a given neuron is nearly the same, between one trial and the next (time scale of a hundred milliseconds)
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Comparison: weak random
Hennequin et al. 2014,
Week 14-part 3: Random Plasticity-optimized circuit
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Random connections, fast
‘distal’ connections, slow, (Branco&Hausser, 2011) structured
12000 excitatory LIF = 200 pools of 60 neurons 3000 inhibitory LIF = 200 pools of 15 neurons
Classic sparse random connectivity (Brunel 2000)
Stabilizy-optimized random connections
Overall:20% connectivity
Fast AMPA
slow NMDA
Week 14-part 3: Stability optimized SPIKING network
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Neuron 1Neuron 2Neuron 3
Spontaneous firing rate
Single neuron different initial conditions
Hennequin et al. 2014
Classic sparse random connectivity (Brunel 2000)
Week 14-part 3: Stability optimized SPIKING network
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Hennequin et al. 2014
Classic sparse random connectivity (Brunel 2000)
Week 14-part 3: Stability optimized SPIKING network
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-Long transients
-Reliable (non-chaotic)
-Rich dynamics (non-trivial)
-Compatible with neural data (excitation/inhibition)
-Plausible plasticity rules
Week 14-part 3: Rich neuronal dynamics: a result list
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Biological Modeling of Neural Networks:
Week 14 – Dynamics and Plasticity
Wulfram GerstnerEPFL, Lausanne, Switzerland
14.1 Reservoir computing - complex brain dynamics - Computing with rich dynamics14.2 Random Networks
- stationary state - chaos
14.3 Stability optimized circuits - application to motor data14.4. Synaptic plasticity - Hebbian - Reward-modulated 14.5. Helping Humans - oscillations - deep brain stimulation
Week 14 – Dynamics and Plasticity
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Hebbian Learning= all inputs/all times are equal
),( postpreFwij
prepost
ij
ijw
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Week 14-part 4: STDP = spike-based Hebbian learning
Pre-before post: potentiation of synapse
Pre-after-post: depression of synapse
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Modulation of Learning= Hebb+ confirmation
confirmation
( , , )ijw F pre post CONFIRM
local global
Functional PostulateUseful for learning the important stuff
Many models (and experiments) of synaptic plasticity do not take into account Neuromodulators.
Except: e.g. Schultz et al. 1997, Wickens 2002, Izhikevich, 2007; Reymann+Frey 2007; Moncada 2007, Pawlak+Kerr 2008; Pawlak et al. 2010
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Consolidation of Learning
Success/reward Confirmation -Novel-Interesting-Rewarding-Surprising
Neuromodulatorsdopmaine/serotonin/Ach
‘write now’ to long-term memory’
Crow (1968), Fregnac et al (2010), Izhikevich (2007)
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Plasticity
BUT - replace by inhibitory plasticity avoids chaotic blow-up of network
avoids blow-up of single neuron (detailed balance) yields stability optimized circuits
Vogels et al.,Science 2011
- here: algorithmically tuned
Stability-optimized curcuits
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Plasticity
BUT
- replace by 3-factor plasticity rules
- here: algorithmically tuned
Readout
Izhikevich, 2007Fremaux et al. 2012 Success signal
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Week 14-part 4: Plasticity modulated by reward
Dopamine encodes success= reward – expected reward
Dopamine-emitting neurons: Schultz et al., 1997
Izhikevich, 2007Fremaux et al. 2012 Success signal
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Week 14-part 4: Plasticity modulated by reward
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Week 14-part 4: STDP = spike-based Hebbian learning
Pre-before post: potentiation of synapse
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Week 14-part 4: Plasticity modulated by reward
STDP with pre-before post: potentiation of synapse
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Quiz: Synpatic plasticity: 2-factor and 3-factor rules
[ ] a Hebbian learning rule depends only on presynaptic and postsynaptic variables, pre and post[ ] a Hebbian learning rule can be written abstractly as [ ] STDP is an example of a Hebbian learning rule
( , , )ijw F pre post CONFIRM
[ ] a 3-factor learning rule can be written as
[ ] a reward-modulated learning rule can be written abstractly as
( , , )ijw F pre post globalfactor
( , , )ijw F pre post success
[ ] a reward-modulated learning rule can be written abstractly as ( , , )ijw F pre post neuroMODULATOR
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Week 14-part 4: from spikes to movement
How can the readouts encode movement?
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Population vector coding of movements
Schwartz et al. 1988
Week 14-part 5: Population vector coding
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•70’000 synapses•1 trial =1 second•Output to trajectories via population vector coding•Single reward at the END of each trial based on
similarity with a target trajectory
Population vector coding of movements
Fremaux et al., J. Neurosci. 2010
Week 14-part 4: Learning movement trajectories
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Per
form
ance
Fremaux et al. J. Neurosci. 2010
QuickTime™ and a decompressor
are needed to see this picture.
R-STDP LTP-only
Week 14-part 4: Learning movement trajectories
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Reward-modulatedSTDP for movement learning - Readout connections, tuned by 3-factor plasticity rule
Fremaux et al. 2012 Success signal
Hebbian STDP - inhibitory connections, tuned by 2-factor STDP, for stabilization
Week 14-part 4: Plasticity can tune the network and readout
Vogels et al. 2011
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Exam: - written exam 17. 06. 2014 from 16:15-19:00 - miniprojects counts 1/3 towards final grade For written exam:
-bring 1 page A5 of own notes/summary-HANDWRITTEN!
Last Lecture TODAY
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Nearly the end: what can I improve for the students next year?
Integrated exercises?
Miniproject?
Overall workload ?(4 credit course = 6hrs per week)
Background/Prerequisites?
-Physics students-SV students-Math students
Quizzes?
Slides? videos?
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Biological Modeling of Neural Networks:
Week 14 – Dynamics and Plasticity
Wulfram GerstnerEPFL, Lausanne, Switzerland
14.1 Reservoir computing - Review:Random Networks - Computing with rich dynamics14.2 Random Networks
- stationary state - chaos
14.3 Stability optimized circuits - application to motor data14.4. Synaptic plasticity - Hebbian - Reward-modulated 14.5. Helping Humans - oscillations - deep brain stimulation
Week 14 – Dynamics and Plasticity
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Week 14-part 5: Oscillations in the brain
Individual neurons fire regularly
two groups alternate
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Week 14-part 5: Oscillations in the brain
Individual neurons fire irregularly
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Week 14-part 5: STDP
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Week 14-part 5: STDP during oscillations
Oscillation near-synchronous spike arrival
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Week 14-part 5: Helping Humans
Pfister and Tass, 2010
big weights
small weights
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Week 14-part 5: Helping Humans: Deep brain stimulation
Parkinson’s disease:Symptom: Tremor
-periodic shaking- 3-6 Hz
Brain activity: - thalamus, basal ganglia- synchronized 3-6Hz in Parkinson-
Deep brain stimulation Benabid et al, 1991, 2009
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Week 14-part 5: Helping Humans
healthy pathologicalAbstractDynamicalSystemsView ofBrain states
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Week 14-part 5: Helping Humans: coordinated reset
Pathological, synchronous
Coordinated reset stimulus
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Week 14-part 5: Helping Humans
Tass et al. 2012
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The EndPlasticity and Neuronal Dynamics
Theoretical concepts can help to understand brain dynamics contribute to understanding plasticity and learning inspire medical approaches can help humans
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now QUESTION SESSION!
Questions to Assistants possible until June 1
The end
… and good luck for the exam!
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Hennequin et al. 2014,
Week 14-part 3: Correlations in Plasticity-optimized circuit