Amitabha Bose Jawaharlal Nehru University amp New Jersey Institute of Technology IISER Pune February 2010 bosejnugmailcom Typical Neuron Applied Mathematicians Neuron Mathematicians Neuron ID: 188287
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Slide1
Introduction to the mathematical modeling of neuronal networks
Amitabha BoseJawaharlal Nehru University & New Jersey Institute of Technology
IISER, Pune February 2010
bose.jnu@gmail.comSlide2
Typical Neuron
Applied Mathematician’s Neuron
Mathematician’s NeuronSlide3
Why are action potentials important?
Action potentials are measurable eventsThe timings or firing rate of action potentials can encode information - place cells in hippocampus
- coincidence detection for sound localization - orientation selectivity in visual cortexNeurons can communicate with one another using action potentialsSlide4
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 neuronSlide5
Crustacean Pyloric Rhythm (CPG)
PD
LP
PY
Hooper 94, 95
Bean, Nature Rev. Neuro. 2007Slide6
Modeling a neuron as an RC circuit
Membrane separates charge
Ions flow through channels causing voltage changesSlide7
Hodgkin-Huxley type equationsSlide8
Outline
Examples of neuronal computationComplications for modelingUnderstanding the Hodgkin-Huxley equationsSimple models to analyze synchronous and anti-phase oscillations
Slide9
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)Slide10
Model for place cell firing(Bose, Booth,Recce 2000)
T
P
I
1
I
2
Inhibition
Excitation
PLACE FIELD
P
= synchronous group of place cellsSlide11
Place cells
Place cells also code for location in 2-dimensional environment. They interact with head direction cells.Slide12
Auditory Cortex
Coincidence 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)Slide13
Model for coincidence detection(Cook et al, 2003, Grande & Spain 2004)
O1
O2
NL
P
O1
O2
Calculate NL firing rate as a function of phase independent of frequency of O
1
and O
2Slide14
Visual Cortex
Neurons fire at preferential orientationsSlide15
Visual Cortex
Very well suited to detecting orientations, contrasts, directions of movement, yet cannot resolve certain visual scenes.
Necker CubeSlide16
Basal Ganglia - Parkinsonian Tremor
Normal state: Irregular, no correlations in STN cellsParkinsonian state: Rhythmic, STN cells cluster
Rubin & Terman, 2004Slide17
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’sSlide18
Lung Episode
Buccal Episode
Overlaid traces show
that the lung bursts begin
at same point in buccal cycle
as in PIR.
Overlaid traces show that the buccal cycles continue predictably from last lung burst.
Ventilatory
Rhythms
Wilson et al 2002
Lung Area
Buccal AreaSlide19
Different types of bursting neurons
(students - ask
Pranay
about this!)Slide20
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 experimentalistsQ. What is the appropriate level of detail for modeling?Q. Is there any fun mathematics to be done?Slide21
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?Slide22
Translation to mathematics
What accounts for the rhythmic activity? Periodic solutions in phase spaceUnder what circumstances do synaptic and intrinsic properties of neurons cooperate or compete?
Effect of parameters on solutionsWhat 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 conditionsSlide23
To the blackboard!