A lecture I gave recently as an introduction to computational neuroscience. Note: the "pentagon of science" was proposed by Eric L Schwartz, a professor at BU who coined the term "computational neuroscience".

1. Introduction to Computational Neuroscience:
Dynamical Systems Modeling
Yohan J. John
Research Scientist
Neural Systems Lab
Boston University

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The Plan
Some “sociology” of neuroscience
What is a computational model?
Intro to differential equations
A simple example of a dynamical system
The “landscape” analogy
The Hodgkin-Huxley model
Other neuroscience applications of dynamical systems
Hands on with some simulations!

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The “Pentagon” of Mind and Brain Sciences

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A spectrum of methodological approaches

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A spectrum of methodological approaches

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A spectrum of methodological approaches

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What is computational modeling?
Mathematical/computational methods are used in two broad approaches:
Statistical modeling
Capturing patterns in data
Uses probabilistic reasoning
• Regression (curve fitting)
• Hypothesis testing (t-tests, ANOVA)
Dynamical modeling
Capturing time-varying processes
Uses deterministic reasoning (in simple cases)
• Differential equations
• “Phase plane” analysis

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What is computational modeling for?
Predicting future (unmeasured) data
Understanding causality
Helping formulate hypotheses for future experiments
“It’s Difficult to Make Predictions, Especially About the
Future”
- Niels Bohr? Yogi Berra? Mark Twain? Nostradamus? Anonymous?

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What is a model?
Mercator projection Lambert equal area projection

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What is a model?
“On Exactitude in Science” (1946)
by Jorge Luis Borges

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Internal (cognitive) models
One of the goals of computational
neuroscience and computational cognitive
science is to understand how humans and
other animals create these internal models.

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What is a model?
"All models are wrong, but some are useful."
-- attributed to statistician George Box
“Useful” can mean different things to
different researchers. Generally, a
model is used to
• simplify the data, capturing the
biggest trends and separating
out “noise” or random
fluctuations
• generate predictions
• explain how the data arose
The geocentric model was useful for
predicting data such as eclipses.
The heliocentric model was initially no
better at prediction than the geocentric
model, but was simpler, provided deeper
insight, and stimulated further theorizing.

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Dynamical Systems
They capture how systems change with respect to time, space, and other variables.
Neuroscience examples:
• Membrane potential (voltage)
• Concentrations of ions (e.g., sodium,
potassium, calcium)
• Firing rate (spikes per second)
• Field potentials (EEG, LFP etc)
• BOLD signals (fMRI)

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Why use dynamical systems?
Differential equations can be more powerful than
statistical fits of specific data sets.
Dynamical systems can help us understand the
similarity between processes that initially seem quite
different.
We will see that dynamical systems can also help us use
quantitative techniques to understand qualitative
properties of systems.
Isaac Newton showed that a cannonball fired fast
enough from a tall mountain could “fall” all the way
around Earth without touching its surface.

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The derivative: capturing rate of change
A derivative (a.k.a. differential) represents
the rate of change of a quantity with respect to time, space, or some other variable.
example: speed is the derivative of distance
with respect to time

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The derivative captures the slope
Slope:

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Differential equations
The simplest
possible differential
equation (other than
zero change!)
example: a car moving at constant speed

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Linear growth
Exact solution:
A constant growth rate

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Exponential growth
Exact solution:
A growing growth rate:
(This is the continuous version of geometric growth.)

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Predator-prey modeling
https://globalchange.umich.edu/globalchange1/current/labs/Lab7_PredatorPrey/Pred_Prey.htm
A simple system… but no simple
(“closed form”) solution!
Coupled differential equations:

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The Lotka-Volterra Predator-Prey Model
A “toy” model with simplifying assumptions, such as:
1. Prey find ample food at all times.
2. The food supply of predators depends only on number of prey.
3. The rate of population growth depends on its size.
4. The environment does not change.
5. Predators have limitless appetite.

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Oscillating populations

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Perturbations to the system
What might happen if there is a
sudden drop in the number of foxes?

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A “paradoxical” perturbation?

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Foxes versus rabbits
We can plot the population of foxes versus
the population of rabbits.
The closed loop indicates the oscillatory
dynamics.
We can also see how the perturbation made
the cycle “bigger”.

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State space
Any collection of variables or
measurements can be understood as
a space.
A specific set of values for each
variable – a possible state of the
system – counts as a point in this
space.
The number of variables is the
number of dimensions of the system.
For 1D, 2D and 3D systems, we can
visualize the state space.

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The “landscape” of a 1D system
Izhikevich, 2007: Dynamical Systems in Neuroscience
The ball represents the state of the system
The initial position of ball could be anywhere
The ball moves downwards according to the
slope (i.e., derivative)
The ball stops at “fixed points”, where slope
is zero
Fixed points can be stable or unstable
stable
stable
unstable

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The “landscape” of a 2D system
With two variables (x and y), we can visualize the
dynamics as a vector field
An arrow represents the rate of change at the
location (x,y)
Arrows point to where a ball would move if placed
at that point
Each diagram of this type is called a “phase plane”
Key concepts: attractors, repellers, saddle points
Izhikevich, 2007: Dynamical Systems in Neuroscience

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https://www.rei.com/learn/expert-advice/topo-maps-how-to-use.html
Unstable points (“repellers”) are
like peaks
Stable points (“attractors”) are
like valleys
Saddle nodes are like ridges
between peaks
The “landscape” of a 2D system

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The “seascape” of a 2D system?
A third kind of structure is
called a limit cycle – it is an
oscillatory pattern
Izhikevich, 2007: Dynamical Systems in Neuroscience

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Foxes versus rabbits on the phase plane
The lengths of the arrows tell us how
large the derivatives are, and therefore
how fast the system will change when
it is at that point.

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The dynamical systems approach
1. Find differential equations that capture how the variables change over time.
2. Study the special features of the system
• Attractors
• Repellers
• Saddle Points
• Limit Cycles
• Strange attractors (a chaotic system!)
3. Use the insights to understand the real system and predict data that haven’t been
collected yet. (This is the hardest step!)
Studying fixed points and other structures of the state space allows us to understand
qualitative features of dynamical systems.

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The neuron: key signaling unit of the brain

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The action potential or “spike”
The action potential travels
along the axon
It is not a current but it involves
currents (which mostly flow
across the membrane)
The axon is not like a metal wire
https://biology.stackexchange.com/questions/34075/how-to-conceptualize-the-action-potential

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The biophysics of resting
All biological cells allow ions to cross their
membranes.
This flow can be passive or active.
Ion channels are selective for specific
ions.
At equilibrium, the inside of the cell is at
a lower potential than the outside.
https://courses.washington.edu/conj/membpot/membranepot.htm

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The biophysics of spiking
Changing the membrane potential with input –
current injection or neurotransmitter binding –
can cause a rapid cascade of events that together
create a spike or action potential.
The key ions involved in this cascade are sodium
and potassium.
Animated videos help understand these events.

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The Hodgkin-Huxley model
Alan Hodgkin and Andrew
Huxley derived a highly accurate
model of this cascade of events
using differential equations.
Schwiening, C. J. (2012). A brief historical perspective: Hodgkin and Huxley. The Journal of
Physiology, 590(11), 2571-2575.
They recorded from the squid
giant axon, and fit their model to
its dynamics.

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The Hodgkin-Huxley model of action potentials
Can you identify which plot
is data and which is the
model?
Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its
application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500-544.

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Dynamics of ion flow and membrane potential
The Hodgkin-Huxley model describes the
dynamics of 4 quantities:
V: membrane potential (voltage)
n: K channel activation
m: Na channel activation
h: Na channel inactivation
m, n and h are “gating” variables that
describe the dynamics of the ion channels

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Simulating a spike using the HH model
n: K channel activation
m: Na channel activation
h: Na channel inactivation

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The Hodgkin-Huxley equations

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The Hodgkin-Huxley legacy
Recall: Hodgkin and Huxley did not know about the structure of ion channels.
“The HH model has been amazingly successful in both describing and predicting a large
number of neuronal properties. Extensions of this model, incorporating a variety of
voltage-dependent channel types beyond the original HH pair, have been very widely used
in research throughout the world. However, as HH were themselves well aware, the
success of the model does not in itself constitute convincing evidence that the "pictorial"
interpretation of the HH equations is a true reflection of the real molecular events. It is
therefore very gratifying, although perhaps surprising, the extent to which modern
investigations into the molecular structure of the various channels have confirmed the
physical reality, or approximate reality, of many aspects of the model.”
Dr W.J. Heitler, University of St Andrews
https://www.st-andrews.ac.uk/~wjh/hh_model_intro/

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Depolarization and hyperpolarization:
the rabbits and foxes of a neuron
The similarity between
the HH model and the
predator-prey model can
be revealed by looking at
the phase plane of the
potassium gate (n) versus
the voltage (V).
For periodic spikes, we see
a limit cycle.
Izhikevich, 2007: Dynamical Systems in Neuroscience

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A single neuron as a dynamical system
Izhikevich, 2007: Dynamical Systems in Neuroscience

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A single neuron as a dynamical system
Izhikevich, 2007: Dynamical Systems in Neuroscience

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A single neuron as a dynamical system
Izhikevich, 2007: Dynamical Systems in Neuroscience

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“Paradoxical” perturbations revisited
Periodic firing – or a limit cycle

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“Paradoxical” perturbations revisited
A transient increase in input disrupted
firing. What happened?

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Dynamical systems used for prediction
John Rinzel and colleagues analyzed
how fixed points and limit cycles
change with changes in parameters
(inputs and initial conditions).
They used this to show that for some
types of neurons, a transient increase
in input can shut down repetitive firing
(!).
They also verified this experimentally.
Izhikevich, 2007: Dynamical Systems in Neuroscience

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Bumping the neuron into rest
Rinzel and colleagues showed that in a type 2
neuron, i.e., one with two stable states, rest and
repetitive firing, it is possible to “bump” the
system from one attractor to the other with a
small increase in input.
Izhikevich, 2007: Dynamical Systems in Neuroscience
Firing
Attractor
Resting
Attractor

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Dynamical systems all the way down (and up!)
The Pyramidal neuron Interneuron Gamma (PING) model
Borgers (2017). An Introduction to Modeling Neuronal Dynamics

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Behavior as an attractor
John, Y. J., Zikopoulos, B., Bullock, D., & Barbas, H. (2018). Visual Attention Deficits in Schizophrenia Can Arise From Inhibitory
Dysfunction in Thalamus or Cortex. Computational Psychiatry, 2, 223-257.

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How the leopard got its spots?
Kondo, S., & Miura, T. (2010). Reaction-diffusion
model as a framework for understanding biological
pattern formation. Science, 329(5999), 1616-1620.

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A “landscape” of memories
Thinking of memories in
terms of dynamical systems
can help us understand their
context-sensitive nature.
A good memory system
needs “deep” and “well-
separated” attractor “valleys”.
A memory system with
shallow valleys can be
bumped into a nearby but
incorrect valley.
Valleys can’t be too
separated… or it will be
difficult to find the “sweet
spot” for memory recall!
http://www.scholarpedia.org/article/Attractor_network
Stephen Grossberg and
John Hopfield were
pioneers of the use of
dynamical systems to
model memory and other
neural processes.

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Memory “landscapes” versus memory “slots”
The dynamical system perspective can help us see the limitations of the
computer metaphor (which is the same as the older library metaphor).

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The persistence of memory
Constantinidis, C., & Klingberg, T. (2016). The neuroscience of working memory capacity
and training. Nature Reviews Neuroscience, 17(7), 438.

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Conclusion:
Why Dynamical Systems?
Dynamical systems methods allow us to model any kind of
time-varying system.
Quantitative methods can be used to gain qualitative
understanding.
The concepts of attractors, repellers, limit cycles etc. can help
us think about seemingly paradoxical phemomena.
For memory, the dynamical systems perspective provides a
useful alternative to the computer metaphor.

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Further reading

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And now… simulations!

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Matlab

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Octave

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Lotka-Volterra: Things to Try
Try different values of alpha, beta, delta
and gamma
Save the file after any editing
To run, press F5
What happens if x = gamma/delta and y
= alpha/beta?

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Hodgkin-Huxley: Things to Try
Change the input strengths
Try the ramping input
Reduce the duration of the
simulation to see the spike
shape more clearly
Change the ion channel
parameters