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