Theoretical neuroscience

PHYSICS 567

Nicolas Brunel (nicolas.brunel@duke.edu)

Practical details

• Problem sets will be issued every Thursday, are due the next Thursday in class.

• Grading will be based on Attendance and participation (10%), homework (40%), and afinal exam (50%)

• Office hours: Friday 3:30pm-5pm, Bryan 101G

Introduction

• Brains, scales, structure, dynamics

• What is computational/theoretical neuroscience?

• Outline of the course

Brain networks: from C Elegans to Humans

Animal # neurons # synapses

C Elegans 302 7,500

Fruit fly 2.5 105 107

Honey bee 106 109

Mouse 108 2 1011

Cat 109 2 1012

Human 1011 2 1014

Spatial scales

∼10cm Whole brain

∼1cm Brain structure/cortical area

100µm- 1mm Local network

10µm- 1mm Neuron

100nm- 1µm Sub-cellular compartment

∼1-10nm Molecule

Experimental tools

The whole brain level: structures

Cerebral cortex: network of areas

Area level

• Areas are interconnected networks of local networks (‘columns’)

Local network level

• Size∼ cubic mm

• Total number of cells∼ 100,000

• Types of cells:– pyramidal cells - excitatory (80%)

– interneurons - inhibitory (20%)

• Total number of synapses ∼ 109

(10,000 per neuron)

• Cells connect potentially to all other celltypes (E→ E, E→ I, I→ E, I→ I)

• Connection probability∼ 10%

Neuron level

• Neuron = complex tree-like structures with many compartments (e.g. dendritic spines)

Subcellular compartment level

• Subcellular compartments (e.g. dendritic spines) contain a huge diversity of molecules(in particular protein kinases and phosphatases) whose interactions define complex

networks

Temporal scales

Days-Years Long-term memory

Seconds-Minutes Short-term (working) memory

100ms - 1s Behavioral time scales/Reaction times

∼ 10ms Single neuron/synaptic time scales

∼ 1ms Action potential duration; local propagation delays

� 1ms Channel opening/closing

Submillisecond

• Molecular time scales (channel opening/closing; diffusion of neurotransmitter insynaptic cleft; etc)

Millisecond

• Width of action potentials; axonal delays in local networks

Tens of ms

• Synaptic decay time constants; membrane time constant of neurons; axonal delays forlong-range connections

Hundreds of ms

• Behavioral time scales (e.g. motor response to a stimulus)

Seconds-minutes

• Short-term memoryWorking memory

Days-years

• Long-term memory

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• Modern Neuroscience generates increasingly large amounts of quantitative data. Needsquantitative models to make sense of this data.

• Art of modeling: Needs proper balance between too little detail and too muchMake things as simple as possible, but not simpler. - A. Einstein

• Model building incorporates biological facts and physical principles

• Useful models make experimentally testable predictions→ stimulate new experiments.

Questions in theoretical/computational neuroscience

What? Describe in a mathematically com-

pact form a set of experimental obser-

vations.

How? Understand how a neural system

produces a given behavior.

Why? Understand why a neural system

performs the way it does, using e.g.

tools from information theory.

Mathematical tools

• Single neuron/synapse models, rate models: systems of coupled differential equations.Tools of dynamical systems (linear stability analysis, bifurcation theory)

• Networks: Graph theory, linear algebra. Large networks: tools of statistical physics

• Noise (ubiquitous at all levels of the nervous system): Statistics, probability theory,stochastic processes.

• Coding: Information theory

Outline

Week 1 Jan 15, 17 Introduction; Membranes, channels

Week 2 Jan 22, 24 Neurons: Hodgkin-Huxley type models

Week 3 Jan 29, 31 Neurons: Leaky integrate-and-fire type models

Week 4 Feb 5, 7 Synapses and synaptic plasticity

Week 5 Feb 12, 14 Networks

Week 6 Feb 19, 21 Network Dynamics: Rate models

Week 7 Feb 26, 28 Network Dynamics: Networks of spiking neurons

Week 8 Mar 5, 7 Coding: Single neurons

Week 9 Mar 19, 21 Coding: Networks

Week 10 Mar 26, 28 Unsupervised learning

Week 11 Apr 2, 4 Supervised and reinforcement learning

Week 12 Apr 9, 11 Learning and memory

Week 13 Apr 16, 18 Computing

Single neuron models

• Hodgkin-Huxley neuron

CdV

dt= −IL(V )−

∑a

Ia(V,ma, ha)+Isyn(t)

IL(V ) = gL(V − VL)

Ia(V,ma, ha) = ga(V − Va)mxaa hyaa

τma (V )dma

dt= −ma +m∞a (V )

τha (V )dha

dt= −ha + h∞a (V )

• Integrate-and-fire neuron

CdV

dt= −gL(V − VL) + Isyn(t)

Fixed threshold Vt, reset Vr , refractory period;

• How do single neurons work?

• What are the mechanisms of action potential generation?

• How do neurons transform synaptic inputs into a train of action potentials?

Single synapse models

• ‘Conductance-based’

Ii,syn(t) =∑

a=E,I

(V − Va)∑j,k

ga,i,jsa(t− tkj )

τaṡa = . . .

• ‘Current-based’

Ii,syn(t) =∑

a=E,I

∑j,k

Ja,i,jsa(t− tkj )

Popular choices of s

– Delayed difference of exponential;

– Delayed delta function

• How do synapses work?

• What are the mechanisms of synapticplasticity on short time scales?

Synaptic plasticity

Synaptic efficacy can be modified in various ways:

• Spike timing(STDP experiments)

• Firing rate(BCM, Sjostrom et al, etc)

• Post-synaptic V(pairing, etc)

• Can we capture this experimental data using simplified ‘plasticity rules’?

• What are the mechanisms of induction of synaptic plasticity?

Networks: architecture

Large scaleShort scale

• How can we model networks at various scales?

Networks: dynamics

• Rate model (neural mass model): de-scribe the activity of a whole population

of neurons by a single ‘average firing

rate’ variable m(x, t);

• Networks of spiking neurons: de-scribe the activity of a population of

N neurons by O(N) coupled differen-

tial equations, coupled through network

connectivity matrix.

Coding

Tools from information theory:

• Shannon mutual information

• Fisher information

• What contains information in single neurons spike trains/population activity?

• How much information is contained in single neurons/population response?

• What is the optimal way of transmitting information about a given input, given itsstatistics and other constraints? Are neural systems optimal/close to optimal at

transmitting information?

Learning and memory

• How do external stimuli triggers changes of neuronal activity in neural circuits?

• How do changes of activity trigger in turn changes in synaptic connectivity?

• How do changes in synaptic connectivity in turn change the dynamics of neuralcircuits?

• How much information can a single neuron/a population of neurons store?

• What kind of learning algorithms/rules allow to reach optimal storage?

Useful books

• Tuckwell, “Introduction to Theoretical Neurobiology”, Vols. I & II (Cambridge U. Press, 1988)

• Amit, “Modeling brain function: The world of attractor neural networks” (Cambridge U. Press, 1989)

• Hertz, Krogh, and Palmer, “Introduction to the Theory of Neural Computation” (Addison-Wesley, 1991 - nowfrom: Perseus Book Group and Westview Press)

• Rieke, Warland, de Ruyter van Steveninck, and Bialek, “Spikes: Exploring the Neural Code” (MIT Press,1997)

• Koch, “Biophysics of Computation: Information Processing in Single Neurons” (Oxford U. Press, 1999)

• Dayan and Abbott, “Theoretical Neuroscience: Computational and Mathematical Modeling of NeuralSystems” (MIT Press, 2001)

• Izhikevich, “Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting” (MIT Press,2007)

• Ermentrout and Terman, “Mathematical Foundations of Neuroscience” (Springer, 2010)

• Gerstner, Kistler, Naud and Paninski “Neuronal Dynamics: From single neurons to networks and models ofcognition” (Cambridge U. Press, 2014)