程序代写代做代考 CompNeuro_CraftofModelling

CompNeuro_CraftofModelling

Dr. Cian O’Donnell
cian.odonnell@bristol.ac.uk

The craft of computational modelling

for neuroscience

COMS30127: Computational Neuroscience

mailto:cian.odonnell@bristol.ac.uk

What we will cover today

• What is a model?

• What is the purpose of computational modelling?

• Levels of abstraction (spatial, temporal and conceptual)

• Compare models of single neurons.

• The Fitzhugh-Nagumo neuron model.

• How should we choose the ‘correct’ model for the problem
at hand?

What is a model?
• A model is a simplified description of a real-world system.

• Models can be:

– Physical (e.g. scale models of buildings)

– Analogical (e.g. billiard-ball model of a gas)

– Phenomenological (e.g. integrate-and-fire neuron)

• Models can be represented by:

– A physical object

– Words

– Mathematical equations

• Overview of the philosophy of models in science:

https://plato.stanford.edu/entries/models-science/

https://commons.wikimedia.org/wiki/File:MathModel.svg

What is a model?

model

What is a computational model?

• Fundamentally, a computational model is just a
mathematical model that is programmed and then
solved or simulated using a computer.

• Technically speaking all computational models are
phenomenological (e.g. Hodgkin and Huxley ignored
quantum mechanics).

• However in practice in neuroscience, most people
consider phenomenological models to be those which
abstract away all laws of (bio)physics.

What is the purpose of a computational model?

“All models are wrong, but some are useful.”

— George Box

What is the purpose of a computational model?

To gain an understanding of a system beyond what we could
achieve via word models alone.

Computational models can be used to:

1. test if a set of concepts are mutually consistent. If not, why?

2. “link levels”, i.e. to ask if a mechanism at one level of
description can account for a phenomenon at another level.

3. simulate experiments that are technically difficult or impossible
to do in the lab.

4. explore “what if?” scenarios that may never occur in the natural
world.

5. validate a formal mathematical analysis.

What could be

What we think might be

What we think is

What actually is

What could be

What we think might be

What we think is

What actually is

1. are these ideas mutually consistent?

?

What could be

What we think might be

What we think is

What actually is

2. can ‘this’ explain ‘that’?

?

What could be

What we think might be

What we think is

What actually is

3. simulate difficult experiments

?

What could be

What we think might be

What we think is

What actually is

4. simulate ‘what if?’ scenarios

?

What is the purpose of a computational model?

Example usages of computational models in neuroscience:

• Hodgkin-Huxley model

(to ask if the squid axon action potential can be explained
by the voltage gating dynamics of sodium and potassium
conductances).

• Simulation of recurrent hippocampal networks with synaptic
plasticity 

(to ask if synaptic plasticity could mediate memory recall
from partial cues).

• Simulating the biophysics of calcium signalling at a synapse
(to explore what happens during synaptic stimulation).

Levels of abstraction

T. Sejnowski

http://cnl.salk.edu/

Spatial Temporal

Action potentialms

s

mins

hours

weeks

years

Neural circuit dynamics

Gene expression

Brain development

Memories

Cellular signalling

Models of single neurons

Abstract Realistic

Abstract models Realistic models
Simple vs Detailed

Hard to relate to biology vs Contains stuff you could measure

Few parameters vs Lots of parameters

Fast simulation vs Slow simulation

Mathematical analysis vs Intractable

Generic vs Specific

Binary
(McCullogh


Pitts)
Firing
rate

Integrate
and
fire

Hodgkin
Huxley Multi-

compartmental
models

Molecular
models

Fitzhugh

Nagumo

The Fitzhugh-Nagumo neuron model

• The Fitzhugh-Nagumo neuron is a reduced
mathematical model of the original HH model
(proposed in 1961-2).

• Its 2D form permits dynamical systems analysis
(much loved by mathematicians).

Neurons as dynamical systems

Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. 

Izhikevich E.M. (2007)

Neurons as dynamical systems

Izhikevich E.M. (2007)

Neurons as dynamical systems

Izhikevich E.M. (2007)

The Fitzhugh-Nagumo model
Consists of two coupled ordinary differential equations for:

1. the voltage V, and

2. the ‘recovery’ variable W.

dV

dt
= V � V 3/3�W + Istim

dW

dt
= 0.08(V + 0.7� 0.8W )

Self-excitation via nonlinear positive feedback

Slower linear negative feedback

The Fitzhugh-Nagumo model

Izhikevich E.M. (2007)

This simple model can recapitulate:

• Appearance of all-or-nothing spike threshold

• Periodic spiking from a constant input current

• Refractory period

• Excitation block

The Fitzhugh-Nagumo model

Prediction of spiking dynamics 

by the Fitzhugh-Nagumo model

FitzHugh, Biophys J (1961)

Prediction of excitation-block

by the Fitzhugh-Nagumo model

http://www.scholarpedia.org/article/FitzHugh-Nagumo_model

http://www.scholarpedia.org/article/FitzHugh-Nagumo_model

This simple model cannot recapitulate:

• Bursting

• Chaotic dynamics

• Type 1 neural dynamics

• The spiking behaviour of many mammalian neurons

As a result, many other dynamical neuron models were
developed (Hindmarsh-Rose, Morris-Lecar, Izhikevich…)

The Fitzhugh-Nagumo model

Which model is best for my problem?

• Choose the form of the model that best matches the granularity
of your scientific question.

• “A model should be as simple as possible, but no simpler” 

— Albert Einstein

• Often this choice is dictated by:

– the data you have to constrain the model

– the phenomenon you wish to explain

– the computational resources you have available

– how much maths/programming you know

– what someone else did previously

End