You can get this book from the Springer site
Calculus For Cognitive Scientists: Derivatives, Integration and Modeling
This book attempts to give a program of self study
on how mathematics, computer science and science can
be usefully and pleasurably intertwined. You must all learn to allow
ideas from mathematics and computation to be part of the way
you approach understanding cognitive and biological science.
A wonderful quote from "Mathematical Models of Social Evolution: A Guide For the
Perplexed" by Richard McElreath and Robert Boyd published by the University of Chicago
Press in 2007 gives you a good perspective on this.
Imagine a field in which nearly all important theory is written in Latin, but most
researchers can barely read latin and certainly cannot speak it. Everyone cites the Latin
papers, and a few even struggle through them for implications. However, most rely upon a tiny cabal
of fluent Latin-speakers to develop and vet important theory.
Things aren't quite so bad as this in evolutionary biology [think our combination
of mathematics, biology and computation here instead!]. Like all good
caricature, it exaggerates but also contains a grain of truth. Most of the important
theory developed in the last 50 years is at least partly in the form of mathematics, the
``Latin'' of the field, yet few animal behaviorists and behavioral ecologists understand
formal evolutionary models. The problem is more acute among those who take
an evolutionary approach to human behavior, as students in anthropology and
psychology are customarily even less numerate than their biology colleagues.
We would like to see the average student and researcher in animal behavior, behavioral
ecology, and evolutionary approaches to human behavior [think biological sciences in general!]
become more fluent in the
``Latin'' of our fields. Increased fluency will help empiricists better appreciate
the power and limits of the theory, and more sophisticated consumers will encourage theorists
to address issues of empirical importance. Both of us teach courses to this end,
and this book arose from our experiences teaching anxious, eager, and hard-working students
the basic tools and results of theoretical evolutionary ecology.
These authors offer much good advice to you too.
You are starting on a new journey where we will try hard to get you
to enjoy this triad of mathematics, computer work and science. So to get the
most out of this class, take the advice of Richard McElreath and Robert Boyd to heart.
While existing theory texts are generally excellent, they often assume
too much mathematical background. Typical students in animal behavior, behavioral
ecology, anthropology or psychology [think biological and cognitive scientists in general!]
had one semester of calculus long ago. It was hard
and didn't seem to have much to do with their interests, and, as a result, they have
forgotten most of it. Their algebra skills have atrophied from disuse, and even
factoring a polynomial is only an ancient memory, as if from a past life.
They have never had proper training in probability or dynamical systems.
In our experience, these students need more hand holding to get them started.
This book does more hand-holding than most, but ultimately learning
mathematical theory is just as hard as learning a foreign language. To this end,
we have some suggestions. In order to get the full use of this book, the reader should
You can get this book from the Springer site
Calculus For Cognitive Scientists: Higher Order Models and Their Analysis
This book again tries to show
how mathematics, computer science and biology can
be usefully and pleasurably intertwined.
The first volume
discussed the necessary one and two variable calculus tools
as well as first order ODE models. In this volume,
we explicitly focus on two variable ODE models both linear and nonlinear
and learn both theoretical and computational tools using MatLab
to help us understand their solutions.
We also go over carefully how to
solve the cable model using separation of variables and Fourier Series.
And we must always caution you to be careful to make sure our use
of mathematics gives us insight.
We also go over carefully how to
solve cable models using separation of variables and Fourier Series.
And we must always caution you to be careful to make sure the use
of mathematics gives you insight. These cautionary
words about the modeling of the physics of stars from 1938
should be taken to heart:
Technical journals are filled with elaborate papers on conditions in the
interiors of model gaseous spheres, but these discussions have, for the most
part, the character of exercises in mathematical physics rather than
astronomical investigations, and it is difficult to judge the degree of
resemblance between the models and actual stars. Differential equations are
like servants in livery: it is honourable to be able to command them, but they
are "yes" men, loyally giving support and amplification to the ideas entrusted
to them by their master. -- Paul W. Merrill, The Nature of Variable Stars,
1938, quoted in Arthur I. Miller Empire of the Stars: Obsession, Friendship,
and Betrayal in the Quest for Black Holes, 2005"
The relevance of this quotation to our pursuits is clear. It is easy to develop
sophisticated mathematical models that abstract from biological complexity
something we can then analyze with mathematics or computational tools in an attempt
to gain insight. But as Merrill says,
it is difficult to judge the degree of
resemblance between the models and actual [biology]
where we have taken the liberty to replace "physics" with our domain here of
"biology". We should never forget the last line
Differential equations are
like servants in livery: it is honourable to be able to command them, but they
are "yes" men, loyally giving support and amplification to the ideas entrusted
to them by their master.
We must always take our modeling results and go back to the scientists
to make sure they retain relevance.
The topics covered here are as follows:
You can get this book from the Springer site
Calculus For Cognitive Scientists: Partial Differential Equation Models
In this volume, we add the new science, mathematics and computational tools
to get you to a good understanding of how excitable neurons generate action potentials.
Hence, we have chapters on some of the basics of cellular processing
and protein transcription and a thorough discussion of how to solve the kinds of linear
partial differential equation models that arise. The equations we use to model
how an action potential develops are derived by making many assumptions about
a great number of interactions inside the cell. We firmly believe that
we can do a better job understanding how to build fast and accurate approximations
for our simulations of neural systems if we know these assumptions and fully understand
why each was made and the domain of operation in which assumptions are valid.
We also spend a lot more time discussing how to develop code
with MatLab/ Octave to solve our problems numerically.
As we have done in the first and second book, we continue to explain our coding
choices in detail so that if you are a beginner in the use of programming techniques,
you continue to grow in understanding so that you can better build simulations of your own.
We always stress our underlying motto: always take the modeling results and
go back to the scientists
to make sure they retain relevance.
Our model of the excitable neuron is based on the cable equation
which is a standard equation in biological information processing.
The cable equation is one example of a linear model phrased in terms
of partial derivatives: hence, it is a linear partial differential equation
model or Linear PDE. We begin by solving the time independent version
of the cable equation which uses standard tools from linear differential equations
in both the abstract infinite length and the finite cable version.
The time dependent solutions to this model are then found using
the separation of variables technique which requires us to
discuss series of functions and Fourier series carefully.
Hence, we have to introduce new mathematical tools that are relevant
to solving linear partial differential equation models. We also discuss
other standard Linear PDE models such as the diffusion equation, wave equation
and Laplace's equation to give you more experience with the separation of variables
technique.
We then use the cable equation to develop a model of the input side
of an excitable neuron -- the dendrite and then we
build a model of the excitable neuron itself. This consists
of a dendrite, a cell body or soma and the axon along which
the action potential moves in the nervous system.
In the process, we therefore
use tools at the interface between
science, mathematics and computer science.
We are firm believers in trying to build models with explanatory power.
Hence, we "abstract" from biological complexity relationships which
then are given mathematical form. This mathematical framework is usually not
amenable to direct solution using calculus and other such tools and hence
part of our solution must include simulations of our model using some sort
of computer language. In this text, we focus on using MatLab/ Octave
as it is a point of view that is easy to get started with.
Even simple models that we build will usually have many parameters that must
be chosen. For example, in the second book, we develop a cancer model
based on a singe tumor suppressor gene having two allele states. There are two
basic types of cell populations: those with no chromosomal instability and other
cells that do have some sort of chromosomal instability due to replication errors
and so forth. The two sets of populations are divided into three mutually exclusive
sets: cells with both alleles intact, cells having only one allele and cells
that have lost both alleles. Once a cell loses both alleles, cancer is inevitable.
Hence, the question is which pathway to cancer is most probable: the pathway where there is
not chromosomal instability or the pathway which does have chromosomal instabilities.
The model therefore has 6 variables -- one for each cell type in each pathway
and a linear model is built which shows how we think cell types turn into other cell types.
The problem is all the two dimensional graphical tools to help us understand such as
phase plane analysis are no longer useful because the full system is six dimensional.
The model can be solved by hand although it is very complicated and it can be
solved using computational tools for each choice of our model's parameters.
If you look at this model in the second book, you'll see there are
4 parameters of interest. If we had 10 possible levels for each parameter,
we would have to solve the model computationally for the 10,000 parameter
choices in order to explore this solution space. This is a formidable task
and it still doesn't answer our question about which pathway to cancer is dominant.
In the second book, we make quite a few reasonable assumptions about how this
model works to reduce the number of parameters to only 2 and then we approximate the
solutions using first order methods to find a relationship between these two
parameters which tells us when each pathway is probably dominant.
This is the answer we seek: not a set of plots but a algebraic relationship between
the parameters which gives insight. Many times these types of questions are the ones
we want to ask.
In this book, we are going to build a model of an excitable neuron and to do
this we will have to make many approximations. At this point, we will have a model
which generates action potentials but we still don't know much about how to
link such excitable neuron models together or how to model second messenger systems
and so forth. Our questions are much larger and more ill-formed than simply generating
a plot of membrane voltage versus time for one excitable neuron. Hence, this entire
book is just a starting point for learning how to model networks of neurons.
In the third book, we develop our understanding further and develop MatLab/ Octave
code to actually build neural circuit models. But that is another story.
The book "Mathematical Models of Social Evolution: A Guide For the
Perplexed" by Richard McElreath and Robert Boyd
also caution modelers to develop models that are appropriately abstract yet simple.
On page 7 - 8, we find a nice statement of this point of view.
This book looks carefully at how to build models to explain social evolution
but their comments are highly relevant to us.
Simple models never come close to capturing all the details in a situation....
Models are like maps -- they are most useful when they contain details of interest
and ignore others....Simple models can aid our understanding of the world in several ways.
There are ... a very large number of possible accounts of any particular biological ... phenomenon.
And since much of the data needed to evaluate these accounts are ... impossible to collect, it can be
challenging to narrow down the field of possibilities. But models which
formalize these accounts tell us which are internally consistent and when conclusions follow
from premises....Formalizing our arguments helps us to understand which [models] are possible
explanations. [Also] formal models are much easier to present and explain. ...They give us the
ability to clearly communicate what we mean. [Further], simple models often lead to surprising results.
If such models always told us what we thought they would, there would be little point in constructing them.
...They take our work in directions we would have missed had we stuck to verbal reasoning, and they
help us understand features of the system of study that were previously mysterious....
Simple formal models can be used to make predictions about natural phenomena.
Further, they make the same important points about the need for abstraction and
the study of the resulting formal models using abstract methods aided by
simulations. On page 8, we have
There is a growing number of modelers who know very little about analytic methods.
Instead these researchers focus on computer simulations of complex systems. When computers
were slow and memory was tight, simulation was not a realistic option. Without analytic methods, it would
have taken years to simulate even moderately complex systems. With the rocketing ascent of
computer speed and plummeting price of hardware,it has become increasingly easy to simulate very
complex systems. This makes it tempting to give up on analytic methods, since most people find
them difficult to learn and to understand.
There are several reasons why simulations are poor substitutes for analytic models.
...Equations -- given the proper training -- really do speak to you. They provide intuitions
about the reasons a ... system behaves as it does, and these reasons can be read from
the expressions that define the dynamics and resting states of the system. Analytic results
"tell us" things that we must infer, often with great difficulty, from simulation results.
This is our point with our comments about the cancer model. Simulation approaches
for the cancer model do not help us find the parameter relationships we seek
to estimate which pathway to cancer is the most likely for a given set
of parameter choices. The analytic reasoning we use based on our dynamic models
helps us do that. In the same way, simulation approaches alone will not
give us insight into the workings of neural systems that can begin to model portions
of the brain circuitry humans and other animals have. The purpose of our modeling
is always to gain insight.
We cover the following material:
You can get this book from the Springer site
BioInformation Processing, A Primer on Computational Cognitive Science
This book tries to show
how mathematics, computer science and science can
be usefully and pleasurably intertwined. Here we begin to
build a general model of cognitive processes in a network
of computational nodes such as neurons using a variety
of tools from mathematics, computational science and
neurobiology. We begin with a derivation of the general solution
of a diffusion model from a low level random walk point of view.
We then show how we can use this idea in solving the cable equation
in a different way. This will enable us to better understand neural
computation approximations. Then we discuss neural systems in
general and introduce a fair amount of neuroscience. We can
then develop many approximations to the first and second messenger systems
that occur in excitable neuron modeling.
Next, we introduce
specialized data for emotional content which will enable us to build
a first attempt at a normal brain model. Then, we introduce tools
that enable us to build graphical models of neural systems in MatLab.
We finish with a simple model of cognitive dysfunction.
We also stress our underlying motto: always take the modeling results and
go back to the scientists
to make sure they retain relevance.
We agree with the original financial modeler's manifesto
due to Emanuel Derman and Paul Wilmott from January 7, 2009
available on the Social Science Research Network which takes
the shape of the following Hippocratic oath. We have changed
one small thing in the list of oaths. In the second item,
we have replaced the original word "value" by the more
generic term "variables of interest" which is a better fit
for our modeling interests.