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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
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
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
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
See Maass et al. 2007
Week 14-part 1: Reservoir computing
-‘calculcate’ 3 4 - if-condition on 1
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
-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
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
I(t)
)(tAn
Week 14-part 2: Review: microscopic vs. macroscopic
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
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
( )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
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
( )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!
( )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
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
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
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
-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
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
Image: Hennequin et al. Neuron, 2014
Re( ) 1 Re( ) 1
Week 14-part 3: Plasticity-optimized circuit
Optimal control of transient dynamics in balanced networkssupports generation of complex movements
Hennequin et al. 2014,
Random stability-optimized circuit (SOC)
Random connectivity
Random
Week 14-part 3: Random Plasticity-optimized circuit
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
slope 1/linear theoryF-I curve of rate neuron
a1= slowest = most amplified
Week 14-part 3: Random Plasticity-optimized circuit
Optimal control of transient dynamics in balanced networkssupports generation of complex movements
Hennequin et al. NEURON 2014,
Churchland et al. 2010/2012 Hennequin et al. 2014
Week 14-part 3: Application to motor cortex: data and model
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)
Comparison: weak random
Hennequin et al. 2014,
Week 14-part 3: Random Plasticity-optimized circuit
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
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
Hennequin et al. 2014
Classic sparse random connectivity (Brunel 2000)
Week 14-part 3: Stability optimized SPIKING network
-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
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
Hebbian Learning= all inputs/all times are equal
),( postpreFwij
prepost
ij
ijw
Week 14-part 4: STDP = spike-based Hebbian learning
Pre-before post: potentiation of synapse
Pre-after-post: depression of synapse
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
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)
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
Plasticity
BUT
- replace by 3-factor plasticity rules
- here: algorithmically tuned
Readout
Izhikevich, 2007Fremaux et al. 2012 Success signal
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
Week 14-part 4: Plasticity modulated by reward
Week 14-part 4: STDP = spike-based Hebbian learning
Pre-before post: potentiation of synapse
Week 14-part 4: Plasticity modulated by reward
STDP with pre-before post: potentiation of synapse
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
Week 14-part 4: from spikes to movement
How can the readouts encode movement?
Population vector coding of movements
Schwartz et al. 1988
Week 14-part 5: Population vector coding
•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
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
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
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
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?
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
Week 14-part 5: Oscillations in the brain
Individual neurons fire regularly
two groups alternate
Week 14-part 5: Oscillations in the brain
Individual neurons fire irregularly
Week 14-part 5: STDP
Week 14-part 5: STDP during oscillations
Oscillation near-synchronous spike arrival
Week 14-part 5: Helping Humans
Pfister and Tass, 2010
big weights
small weights
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
Week 14-part 5: Helping Humans
healthy pathologicalAbstractDynamicalSystemsView ofBrain states
Week 14-part 5: Helping Humans: coordinated reset
Pathological, synchronous
Coordinated reset stimulus
Week 14-part 5: Helping Humans
Tass et al. 2012
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
now QUESTION SESSION!
Questions to Assistants possible until June 1
The end
… and good luck for the exam!
Hennequin et al. 2014,
Week 14-part 3: Correlations in Plasticity-optimized circuit