Introduction to Modeling and
Computational Neuroscience
using Python
I wonder why. I wonder why.
I wonder why I wonder.
I wonder why I wonder why
I wonder why I wonder! *
Randy Heiland
Research Scientist
Indiana University
* Surely You're Joking, Mr. Feynman! (Adventures of a Curious Character), Richard Feynman.
Overview
• Computational Science:
Experiments, Models, Simulations, Analyses
• Modeling
• Neuroscience
• Python programming
• Questions? (+ short survey)
0.00000000e+00,
7.59139762e-07,
1.39243600e-07, 3.508636 5.32143167e- 512 electrode chip
Courtesy A.Litke
05, 2.44901501
1.73985789e-08, 0.00000000e+00
1.45115517e-06, 4.93031343e-09,
3.11438363e-08, 1.92049751e-06,
2.16824470e-08, 6.40914027e-09,
2.65017478e-06, 7.59736245e-07,
1.45118790e-06, 0.00000000e+00,
3.30347955e-06, 7.32821449e-08,
, 7.22468653e-06,
1.50957302e-08,
1.05983595e-06,
2.70174543e-06, 4.93023466e-09,
1.16339858e-08, 0.00000000e+00,
4.09147792e-08, 5.44135943e-07,
2.33556551e-05, 8.26447814e-09,
3.32418446e-06,
[ 2.56281682e-07, 3.11322475e-08,
7.34055651e-08, 4.09002090e-08,
0.00000000e+00, 3.82542228e-09,
Mental
Experiment
Analysis
 Patterns,
Structures,
Causality…
Model
u’ = f(u,v)
v’ = g(u,v)
Mathematical
Simulation
Procedural
Courtesy of Indiana University
Modeling
• Using math to approximate some process (or data)
(physical, biological, chemical, social, …)
• Typically ignore some parameters for a more
“reasonable” model (e.g., for your F=ma lab, you
ignored friction)
• “ all models are wrong, but some are useful”
– Prof. Emer. George E. P. Box, Statistics, UW-Madison
Simulation
•
•
•
•
Execution of a model in a computer program
Several computer languages: C/C++, Java, etc.
Higher level languages: MATLAB, Python, etc.
Solving some models may require the use of
parallel computing.
Python language
•
•
•
•
•
>>> (52*55)/3.0
953.3333333333334
>>> from math import *
>>> cos(pi)
-1.0
Easy to use
Interpreted
Powerful
Free (“open source”); runs on all computers
Used in many computational science tools
Simple model in Python
Average monthly temperature
in Indiana (from weather.com)
 A function as a model
lowTemp=[19,22,30,41,52,61,65,63,55,43,34,23]
plot(lowTemp,'o')
# ‘o’  circular points
time = arange(0, 12, 0.1)
F = 22*sin(pi/6 * (time-pi)) + 42
plot(time,F)
(Note that Python is “0-based indexing”)
Neuroscience
• Goal: understand relationship between neural
structures and functions
Why?
• Help solve important problems
– Vision
– Memory
– Auditory
…
• Ponder/explain interesting topics
– Consciousness; self-awareness
– Altruism
Concepts & Terminology
Soma
A neuron is a nerve cell.
It contains a soma, dendrites,
and axon.
(Combination of biology, chemistry,
and physics)
Illustrative image
Microscopic image
Yes, it’s quite complex
Soma
SEM image: cutaway of (mouse)
nerve ending and its synaptic
vesicles
(cellimagelibrary.org)
The scale of a model
• Models can be at different scales:
molecular, cellular, multi-cellular, tissue,
organ, organism, …
• A single neuron model is on a different space
and time (“spatiotemporal”) scale than a
brain region or whole brain model.
Larger number of neurons
 complex signaling and networks
Rich-Club Organization of the Human Connectome.
M.P. van den Heuvel and O. Sporns. J. Neuroscience,
2 Nov 2011.
100 billion neurons in human brain
(Thanks to Journal of Neuroscience and authors for permission
to use this image)
• “… it is intolerable that we do not have this
information [connectional map] for the
human brain. Without it there is little hope of
understanding how our brains work…”
(Crick 1993)
Q: who knows who Francis Crick was?
Hint: James Watson, Maurice Wilkins, Rosalind Franklin
Francis Crick, 1916-2004
Vision: conversion of light to
electrical signals
Memory
• One of the main hypotheses in neuroscience is
that memories are encoded in the strengths of
synapses between neurons
Soma
• Plasticity – the ability to
change as a result of experience
Overwhelmed yet?
• Let’s begin by modeling a single neuron
• How?
– An electrical circuit is a good starting model
• Why?
– Because it is an (chemo)electrical circuit
From your past homework
The nervous system of the human body contains
axons whose membranes act as small
capacitors. A membrane is capable of storing
1.2 x 10-9 C of charge across a potential
difference of 0.07 V before discharging nerve
impulses through the body. What is the
capacitance of one of these axon membranes?
 neuron spike
Single neuron simulation
http://phet.colorado.edu/en/simulation/neuron - play the simulation to see V flip/spike
Simplified model of a neuron
RC circuit (Resistance-Capacitance)
Ion transports
act as a resistor.
Cell membrane acts as a
capacitor.
http://phet.colorado.edu/en/simulation/circuit-construction-kit-ac-virtual-lab
Leaky integrate-and-fire (LIF) model (#1)
dV 1
= (-V + I × R)
dt RC
Where variables refer to membrane’s:
V = potential
R = resistance
I = current
if t > t_rest; otherwise = 0
• Note: for those who haven’t taken Calculus (yet),
dV/dt is a derivative (not division)
• This equation is a differential equation
• It can be solved numerically
LIF in Python
# initialize all variables
…
for i, t in enumerate(time): # loop over desired time
if t > t_rest:
V[i] = V[i-1] + (-V[i-1] + I*R) / tau * dt
if V[i] >= V_threshold:
V[i] = V[i] + V_spike
t_rest = t + tau_ref
plot(time,V)
Numerical integration via
forward Euler method.
(Rf. www.neurdon.com/tag/spiking-neurons)
But it’s not very realistic…
dV 1
= (-V + I × R)
dt RC
Hodgkin-Huxley model (#2)
• 1952 (Nobel Prize 1963)
Giant squid  measure voltages  Circuit model
• Electrical circuit model
• A more complicated differential equation
- has both linear and nonlinear components
What about modeling LOTS of neurons
• To model a brain, we want to model a network
of neurons
• Need models of both the spiking behavior and
the synapse (transmission between neurons)
In the human brain, each neuron is
connected to several thousand
other neurons.
http://connectomethebook.com/?page_id=58#all
Izhikevich model (#3)
dv
2
= 0.04v + 5v +140 -u+ I
dt
du
= a(bv -u)
dt
v=c
if v >= 30 mV then
u=u+d
• m=membrane potential; u=membrane recovery
• Captures multiple types of spiking behavior
• Computationally fast enough to do many neurons
Izhikevich model (cont’d)
(plus many more)
• Izhikevich E.M. (2003) Simple Model of Spiking
Neurons. IEEE Transactions on Neural Networks,
14:1569- 1572
Analysis
• After you have a model and have run a
simulation of the model, you need to analyze
the resulting data. Similarly, data from an
experiment needs to be analyzed.
– Transfer Entropy is just one such analysis technique
(next slide)
• The Fourth Paradigm: Data-Intensive Scientific
Discovery (2009)
– http://research.microsoft.com/en-us/collaboration/fourthparadigm
Visualization of analysis results
Transfer Entropy (TE) measures the effect that
one neuron’s spiking has on another neuron.
max
min
Part of much larger TE matrix
Graph displays of TE
(with different thresholds)
DIY neuroscience experiments
www.backyardbrains.com (SpikerBox: $50-$100)
♪
input
output
http://www.youtube.com/watch?v=edEXKiOmPvE
Cockroach leg
Sampling of Research[ers]
John Beggs, Physics,
Indiana U.
- experimental research
(“brains in a dish”) and
analysis of experimental
and simulated data
Nancy Kopell, Mathematics,
Boston U.
Co-Director of Center for
Biodynamics
- develops models for and
analysis of networks of
neurons, esp. rhythms and
oscillations.
Susan Amara,
Neurobiology,
U. of Pittsburgh,
past President of Society
for Neuroscience.
- molecular & cellular
biology of transporters
Olaf Sporns,
Psychological and Brain
Sciences, Indiana U.
- computational
cognitive neuroscience;
neural networks; coined
“Connectome”
Some related textbooks
Summary
• Modeling uses math to approximate reality
• Modeling occurs at multiple scales
• Neuroscience: understand relationship
between neural structures and functions
• Python lets you experiment with
computational neuroscience (for free)
IU might be of interest to you
• Dept of Physics
– Biophysics, Biocomplexity
• School of Informatics and Computing
– Complex systems, computer science, data mining
• Dept of Pyschological and Brain Sciences
– Learning, computational models of …
• Biology, Chemistry, …
Thanks!
Be curious. Be creative.
Be nice to your neurons.
Questions & short survey
Further reading/References
•
•
•
•
•
•
•
•
http://www.neurdon.com/tag/spiking-neurons/
http://connectomethebook.com/
Vase or Face? A Neural Correlate of Shape-Selective Grouping Processes in the
Human Brain. Journal of Cognitive Neuroscience, Aug 2001, pg 744-753
http://www.scientificamerican.com/article.cfm?id=is-it-possible-to-use-more
http://blogs.scientificamerican.com/streams-of-consciousness/2011/09/28/goldiehawn-plunges-into-brain-science/
http://faculty.washington.edu/chudler/what.html
http://www.symmetrymagazine.org/cms/?pid=1000591 - From Eye to Sight, Alan
Litke
Craddock TJA , Tuszynski JA , Hameroff S (2012) Cytoskeletal Signaling: Is Memory
Encoded in Microtubule Lattices by CaMKII Phosphorylation? PLoS Comput Biol
8(3): e1002421. doi:10.1371/journal.pcbi.1002421
http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.100
2421
Analysis  publications
• Simple model of spiking neurons.
IEEE Transactions on Neural Networks (2003) 14:1569- 1572
• Complex network measures of brain connectivity: uses and
interpretations.
Neuroimage. 2010 Sep;52(3):1059-69
• Extending Transfer Entropy Improves Identification of
Effective Connectivity in a Spiking Cortical Network Model
Neuroimage. 2010 Sep;52(3):1059-69
Self-awareness
• Awareness of one’s own ability to think
(humans, apes, dolphins, …)
• 1970 “mirror test” for chimpanzees
– Difficult to test
I wonder why. I wonder why.
I wonder why I wonder.
I wonder why I wonder why
I wonder why I wonder! *
• Effects of meditation experience on
functional connectivity of distributed brain
networks. Feb 2012.
www.frontiersin.org/human_neuroscience/10.3389/fnhum.2012.00038/abstract
“Participants with more meditation experience exhibited increased connectivity within
attentional networks…”
• Nerve cells are formed during fetal life and for a
short time after birth. Unlike most cells, which
have a fairly short lifespan, neurons in the brain
live a long time. These cells can live for up to 100
years or longer. To stay healthy, living neurons
must constantly maintain and repair themselves.
• Groups of neurons in the brain have special jobs.
For example, some are involved with thinking,
learning, and memory. Others are responsible for
receiving information from the sensory organs
(such as the eyes and ears) or the skin. Still others
communicate with muscles, stimulating them
into action.
Definitions (for this talk)
• Model: math equation(s) to describe a
process (physics,chemistry,biology,…)
• Simulation: computer program to solve
the model
• Analysis: interpretation/verification of data
from the simulation (or experiment)
Is it possible to use more of our brain?
Barry Gordon, professor of neurology and cognitive science at the Johns Hopkins University School of Medicine, replies
Scientific American| Saturday, March 3, 2012 | 2
Yes! Though perhaps not how you might imagine. You can't put more of your brain to work. Your whole
brain is working all the time, even when you think you're just being lazy. What you can do is make it work
more productively.
There are two proved strategies to make your neural systems more efficient. The first strategy is to focus,
which is hard to do. It is quite difficult to force your brain to stay on task and to shut off extraneous
thoughts. Yet by concentrating, your brain can muster the neural tools it needs to tackle a complex
problem. In fact, intense focus may be one reason why so-called savants become so extraordinary at
performing extensive calculations or remembering a slew of facts.
The second approach is optimization. The human brain is far from an ideal "thinking machine." Our
mental processes are slow, and the accuracy of our memory is far from perfect. Our intrinsic limitations
are compounded by the simple mental blunders we make; these unhelpful tendencies, however, are
correctable. For instance, you can become a better problem solver by looking beyond your personal
biases and blind spots to consider alternative solutions. The more you learn to recognize and seek a
variety of answers, the better your brain will be at finding optimal solutions.
Some proof that focus and optimization can improve the brain's performance comes from research on
video gamers. Neuroscientists at the University of Rochester have shown that even novice gamers can
improve cognitive skills such as perception and attention by playing action video games. These games
can strengthen players' mental acuity because they require intense concentration and ruthless selfcorrection (otherwise, your friends shoot you!).
Sometimes, however, you may think better when you're not trying so hard. (You have to consider all the
alternatives.) Periods of artistic and scientific creativity—when people often tackle the biggest, most
open-ended problems—usually require letting your brain meander, percolate, chill. It may not feel like
you are using more of your brain when you unleash it in this way, but one virtue of the human brain is
that it often does its best work when it does not seem to be working at all.
1-min lesson in Diff. Eq.
• Geometric interpretation
• Show sin(x), tangent/deriv  cos(x)
• d/dt F(x) = cos(x)
This confocal micrograph shows specialized cells named Purkinje cells (red) that are
found in a part of the brain called the cerebellum. They send out vast numbers of
branches that make connections with other cells in the cerebellum. This part of the
brain co-ordinates your voluntary movements and keeps you oriented in space. It also
plays a part in learning physical skills - like riding a bike or playing the piano.