Introduction to the mathematical modeling of neuronal networks

Amitabha Bose

Jawaharlal Nehru University &

New Jersey Institute of Technology

IISER, Pune February 2010bose.jnu@gmail.com

Typical Neuron

Applied Mathematician’s

NeuronMathematician’s

Neuron

nRx

xfx

)('

Why are action potentials important?

• Action potentials are measurable events• The timings or firing rate of action potentials

can encode information

- place cells in hippocampus

- coincidence detection for sound localization

- orientation selectivity in visual cortex

• Neurons can communicate with one another using action potentials

Synaptic Communication

• An action potential in the pre-synaptic neuron provides a current to the post-synaptic cell.

• The effect may be either excitatory (naively thought to promote firing of post-synaptic neuron) or inhibitory (naively having the opposite effect).

• Synapses have their own time scales for rise and decay.• Synaptic delays may be involved.• Synapses can change strength as a function of usage;

this is called synaptic plasticity.

Pre-synaptic neuron Post-synaptic neuron

Crustacean Pyloric Rhythm (CPG)

PD

LP PY

Hooper 94, 95Bean, Nature Rev. Neuro. 2007

Modeling a neuron as an RC circuit

• Membrane separates charge• Ions flow through channels

causing voltage changes

m

m

I I

, /

/ , 1/

I

c i

m

i m m

m

I

Q C V Ic dQ dt

I V R R g

dVC gVdt

Hodgkin-Huxley type equations,( , )

( )

( )

( )

( )

app ion i i

i i

m

i i

h

dvC I I v m hdt

dm m v m

dt v

dh h v h

dt v

( ) ( )[ ]

( ) ( )[ ]

[ ]

Na

K

p qNa NaNa Na

r sK KK K

L L L

I g m v h v v V

I g m v h v v V

I g v V

Outline

• Examples of neuronal computation• Complications for modeling• Understanding the Hodgkin-Huxley

equations• Simple models to analyze synchronous

and anti-phase oscillations

Place cells• Pyramidal cells in hippocampus fire only when animal is

in a specific, known location (transient & stable)• Uses visual cues to trigger memory recall • O’Keefe (1971)

Model for place cell firing(Bose, Booth,Recce 2000)

T P

I1 I2

Inhibition

Excitation

PLACE FIELD

P = synchronous group of place cells

Place cells• Place cells also code for location in 2-dimensional

environment. They interact with head direction cells.

Auditory CortexCoincidence detection

• Neurons have higher firing rate when they get coincidental input from left and right ears

• Owls use this to locate prey and prey to locate owls

• Jeffress delay line model for barn owls (1948)

Model for coincidence detection(Cook et al, 2003, Grande & Spain 2004)

O1 O2

NL

P

t

t

P

O1 O2

Calculate NL firing rate as a function of phase independent of frequency of O1 and O2

Visual Cortex• Neurons fire at preferential orientations

Visual Cortex

• Very well suited to detecting orientations, contrasts, directions of movement, yet cannot resolve certain visual scenes.

Necker Cube

Basal Ganglia - Parkinsonian Tremor• Normal state: Irregular, no correlations in STN cells• Parkinsonian state: Rhythmic, STN cells cluster

Rubin & Terman, 2004

Central Pattern Generator• Important for coordinated movement of muscle groups. • Rhythmic and very stable behavior

Crustacean Pyloric Rhythm (CPG)

PD

LP PY

Nadim et al, 2000’s

Lung EpisodeBuccal Episode

Overlaid traces show that the lung bursts beginat same point in buccal cycleas in PIR.

Overlaid traces show that the buccal cycles continue predictably from last lung burst.

Ventilatory RhythmsWilson et al 2002

Lung Area

Buccal Area

Different types of bursting neurons (students - ask Pranay about this!)

Modelers Goldmine or Minefield? (title borrowed from K.B. Sinha)

• Many complications: high dimensionality, multiple time scales, stochasticity, noise, large networks, unknown architecture…

• Many advantages: “young field”, no canonical equations thus much freedom, growing number of interactions with experimentalists

Q. What is the appropriate level of detail for modeling?

Q. Is there any fun mathematics to be done?

General Biological Questions

• What accounts for the rhythmic activity?• Under what circumstances do synaptic

and intrinsic properties of neurons cooperate or compete?

• What effect do multiple time scales have?• What are the underlying neural

mechanisms that govern behavior?

Translation to mathematics

• What accounts for the rhythmic activity?

Periodic solutions in phase space• Under what circumstances do synaptic and intrinsic

properties of neurons cooperate or compete?

Effect of parameters on solutions• What effect do multiple time scales have?

Singular perturbation theory• What are the underlying neural mechanisms that govern

behavior?

Deriving mathematically minimal models that reveal

necessary and sufficient conditions

To the blackboard!