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Theoretical Neuroscience: Correlation structure of neuronal networks
(summer term 2013)
Description:advanced course for
 MS Physics (minorfield module "Biophysics")
 PhD students in Computational Neuroscience
This lecture is part of
the module "Biophysics" of the physics master program, but it is open
to students at the bachelor level and students from other domains as
well. After an overview and introduction into the field we will
systematically develop the theory of fluctuations and correlations in
neuronal networks. This active topic of current research addresses the
foundations of many aspects of network function like plasticity and
learning and will serve us to introduce the prominent neuronal network
models and theoretical tools presently in use.
Starting from linear OrnsteinUhlenbeck processes
(Langevin equations), over the classical binary (spin) network model to
the spiking leaky integrateandfire model, the defacto standard of
contemporary computational neuroscience, we will develop the methods
required to gain an analytical understanding of network dynamics. The
sequence of models from low to high realism and corresponding
analytical complexity will allow us to first clarify the concepts, and
subsequently treat the analytically more challenging topics.
The student will get acquainted with elementary methods
from linear system's theory, Fourier methods, point processes, as well
as methods from statistical mechanics (Markov processes, master
equation, ChapmanKolmogorov equation, noise and diffusion processes,
FokkerPlanck equation and nonequilibrium steady states) as they are
applied in theoretical neuroscience.
The neuroscientific topics will include the meanfield
theory for binary and spiking networks, the balanced state in
nonspiking and spiking networks, the mechanisms of decorrelation and
of oscillations in recurrent cortical networks, the classical
meanfield theory of pairwise correlations in binary (spin) networks,
the corresponding stateoftheart theory for spiking networks,
including the temporal structure of correlations. We will end with an
outlook on probabilistic inference and nonequilibrium properties of
neuronal networks.
The weekly lecture (45 minutes) is accompanied by
exercises (90 minutes) to discuss the homework. Homeworks will consist
of a mixture of theoretical exercises and (simple) programming and
simulation exercises (preferably using python/numpy/scipy/matplotlib)
to deepen the topics of the lecture.
Contents:1. Course introduction
 What is theoretical Neuroscience?
 Why cortex?
 Why correlations?
 Properties of cortical networks
 Impulse response and transfer function of the leaky integrateandfire (LIF) neuron
2. Pairwise correlations in spike trains
 pairwise correlations in time and frequency domain
 spiketrain correlations
 shotnoise correlations
 correlations in population signals
 WienerKhinchin theorem
 coherence between two deterministic signals
 autocorrelation of a stationary Poisson process
3. Theory of correlations in linear rate models
(Moritz Helias)
 rate modulated Poisson processes
 equivalent fluctuating rate dynamics
 definition of rate models
 solution of the rate dynamics with output noise
 populationaveraged covariances
 explicit calculation of the cross covariances
 backtransform to time domain
Exercise
(Moritz Helias)
 OrnsteinUhlenbeck process
 Dichotomous and Gaussian white noise
 population average of uncorrelated noise
 explicit form of the cross spectrum for an EI network
 explicit form in time domain
4. Meanfield theory of binary networks
(Moritz Helias)
 master equation for binary neurons
 time evolution for the first moment
 meanfield solution
 stability and response of the mean activity
Exercise: Attractor network
(Moritz Helias)
 ground state without synaptic fluctuations
 ground state with synaptic fluctuations
 numerical check
 embedding a cellassembly
 local stability of the ground state
 appearance of the second attractor
5. Theory of pairwise correlations in binary networks
(Moritz Helias)
 master equation for the joint probability distribution, Markov property
 singleneuron susceptibility and linearized equation for correlations
 example EI network
 suppression of fluctuations and correlations
 temporal structure of correlations
 equivalence of binary neurons and linear rate model
Exercise: Correlations in binary networks
(Moritz Helias)
 Correlation caused by a single synapse
 fluctuations in an inhibitory network
 network susceptibility
6. Timeresolved covariance functions for binary neurons
(Moritz Helias)
 equivalence of binary neurons and linear rate model
Exercise: Correlations in binary networks
(Moritz Helias)
 projection solution of covariance matrix
 network susceptibility
7. Spiking neurons
(Moritz Helias)
 subthreshold dynamics
 derivation of the FokkerPlanck equation
 application to the leaky integrateandfire model
 stationary solution
Exercise: Neuron dynamics as diffusion equations
(Moritz Helias)
 neuron driven by Gaussian white noise
 the PIF model
8. Linearresponse theory for spiking neurons
 linearresponse theory
 effective coupling strength for the LIF model
 PIF model
 integral impulse response
 integral linear response of the LIF model
9. Decorrelation of neuralnetwork activity by inhbitory feedback
 correlation suppression in leakyintegrateandfire (LIF) networks
 linearised network dynamics
 population averaged dynamics
 inhibitory networks
 excitatoryinhibitory networks (Schur decomposition)
 population averaged correlations
 variability of a linear decoder
 populationaveraged linear dynamics for an inhibitory random network
10. Meanfield theory and oscillations in LIF networks
(Moritz Helias)
 meanfield description of spiking networks
 the LIF model: the harmonic oscillator of neuroscience
 perturbative treatment of the timedependent FokkerPlanck equation
 homogeneous solution
 boundary condition for the function values
 flux boundary conditions (derivative)
 transfer function
 notes on Hermiticity
Literature:
 Risken, The FokkerPlanck equation (excerpts), Springer
 Ginzburg & Sompolinsky (1994), Theory of correlations in stochastic neural networks, Phys Rev E 50(4):3171–3191
 van Vreeswijk & Sompolinsky (1998), Chaotic balanced state in a model of cortical circuits, Neural Comput 10:1321–1371
 Brunel (2000), Dynamics of sparsely connected
networks of excitatory and inhibitory spiking neurons, J Comput
Neurosci 8(3):183–208
 Renart et al. (2010), The asynchronous state in cortical cicuits, Science 327:587–590
 Tetzlaff et al. (2012), Decorrelation of neuralnetwork activity by inhibitory feedback, PLoS Comput Biol 8(8):e1002596
 Helias et al. (2013), Echoes in correlated neural systems, New J Phys 15:023002
 see also references in lecture material
Additional Information:
 SWS: 3
 ECTS credits: 5 ECT (after passing written exam and participation in exercises [protocols, oral presentations])
 language: English
 prerequisites: background in mathematics equivalent to bachelorlevel in physics recommended
 location: RWTH Aachen University, department of Physics, room 26C 401
 time: summer term 2013, Thursday's, 4am6:30pm (starting April 11th)
 exam: written exam at the end of the course

