Mathematical modeling
applied to neuroscience
Amitabha Bose
Department of Mathematical Sciences
New Jersey Institute of Technology
Hunter College High School 2013
Mathematics and the Natural World
• “The Unreasonable Effectiveness of Mathematics in
the Natural Sciences” published in 1960 by Nobel
Prize winning Physicist Eugene Wigner
• Gives numerous examples from physics like Newton’s
gravitational laws or Einstein’s theory of relativity
• “How can it be that mathematics, being after all a
product of human thought which is independent of
experience, is so admirably appropriate to the
objects of reality?” — Albert Einstein
Mathematics and Life Sciences
• Wigner concludes with “A much more difficult
and confusing situation would arise if we could,
some day, establish a theory of the phenomena
of consciousness, or of biology, which would be
as coherent and convincing as our present
theories of the inanimate world.”
• “There is only one thing which is more
unreasonable than the unreasonable
effectiveness of mathematics in physics, and this
is the unreasonable ineffectiveness of
mathematics in biology.” — Israel Gelfand
Mathematical Physiology
• Many successful applications of mathematics to “solving”
biological problems
• Derivation of the Hodgkin-Huxley equations (1952)
• FitzHugh-Nagumo equation (1962)
• Understanding cardiac dynamics (Peskin 1970’s on, Keener
1980’s on)
• Understanding dynamics of pancreatic beta cells associated
with diabetes (Miura 1970’s, Sherman 1980’s)
• Neuroscience – Rinzel, Wilson-Cowan 1970’s on, Kopell,
Ermentrout 1980’s on, Terman 1990’s on
• Great book by Art Winfree “The Geometry of Biological
Time (1980)
• Other important texts by Jim Murray, Keener & Snyed,
Hoppensteadt & Peskin, Edelstein-Keshet
Outline
•
•
Basics of neuroscience, circuit theory and
differential equations
Numerous applications of math & neuroscience
Q. What is the appropriate level of detail for modeling?
Q. Is there any fun mathematics to be done?
Typical Neuron
Applied
Mathematician’s
Neuron
Mathematician’s
Neuron
x'  f ( x)
x  Rn
What is a derivative?
• A mathematical object that measures the rate
of change of a quantity
• Ex: the slope of a line measures the rate of
change of the “rise” to the “run”.
• Ex: think of the formula s = v·t to measure
distance s. The change in distance per unit
time is the velocity. As the unit of time is
made smaller, this rate becomes the derivative
ds/dt = v.
What is a differential equation?
• An equation composed of derivatives and
functions of the dependent and independent
variables.
2
d

d
• Ex: Damped Pendulum


 sin   0
2
dt
dt
• Differential equations can be “solved” by
analytically or numerically integrating them
for a given set of initial conditions.
Modeling a neuron as an RC circuit
• Membrane separates charge
• Ions flow through channels
causing voltage changes
Im  Ic  Ii ,
dQ
Q  C mV ,
 Ic
dt
dV
I  current
 Cm
 Ic
dt
Q  charge
1
V  IiRm, g 
V  voltage
Rm
R  resistance
dV
C  capacitance Cm
 Im  gV
dt
Hodgkin-Huxley type equations
dv
C
 Iapp   Iion (v, mi , hi )
dt
dmi m (v )  mi

dt
 m (v )
dhi h (v )  hi

dt
 h (v )
INa  g Na mNa p (v)hNa q (v)[v  VNa]
IK  g K mK (v)hK (v)[v  VK ]
r
IL  gL[v  VL]
s
Why are action potentials important?
•
•
Action potentials are measurable events
The timings or firing rate of action potentials can
encode information
- orientation selectivity in visual cortex
- coincidence detection for sound localization
- place cells in hippocampus
•
Neurons can communicate with one another using
action potentials via synapses or gap junctions.
Crustacean Pyloric Rhythm (CPG)
PD
LP
Bean, Nature Rev. Neuro. 2007
PY
Nadim et al,
2000’s
Visual Cortex
• Neurons fire at preferential orientations
Hubel and Wiesel 1962 and related work
Many mathematical models for describing this phenomenon
The brain is good at detecting edges, but not so good at
other things…
Count the black dots
What kind of mathematical model can explain this?
Perceptual Bistability
Rinzel and collaborators are
developing mathematical
models to explain bistability
Auditory Cortex
Coincidence detection
• Neurons have higher firing
rate when they get
coincidental input from left
and right ears
• Owls use this to locate prey
and prey to locate owls
• Jeffress delay line model for
barn owls (1948)
Model for coincidence detection
(Cook et al, 2003, Grande & Spain 2004)
O1 O2
O1
P
O2
t
NL
t
 
P
Calculate NL firing rate as a function of phase
independent of frequency of O1 and O2
Place cells
• Pyramidal cells in hippocampus fire only when animal is in a
specific, known location (transient & stable)
• Uses visual cues to trigger memory recall
• O’Keefe (1971)
Model for place cell firing
(Bose, Booth,Recce 2000)
PLACE FIELD
T
P
I1
I2
P
Inhibition
Excitation
= Place cells
Model Predictions
• Location within place field
• Length of place field
• Behavior in running wheel
Parkinson’s Disease
• Believed to be a disorder of the Basal Ganglia
• Results in tremor, uncontrollable motions,
inability to begin movement
• No experimental consensus on what causes
PD or how it can be treated
• Mathematicians have gotten involved with
experimentalist to figure out the underlying
neural mechanisms
• Peter Tass’ group (Germany)
• Terman, Rubin, Wilson (US)
• Kopell, Wilson (US)
PD results when the output from
Gpi/SNr becomes too synchronized
I
For severe cases, Deep Brain Stimulation has shown remarkable promise for alleviating
PD symptons. We don’t know why DBS works?!
Rubin & Terman (2004) proposed that DBS targets STN
• Normal state: Irregular,
no correlations in STN
cells
• Parkinsonian state:
Rhythmic, STN cells
cluster
Modeling Sleep Rhythms
• Your brain is very active when you are asleep!
• The function of sleep is still not totally
understood, but it is important for learning,
for the immune system, for growth…
• REM sleep – Rapid Eye Movement sleep
(dreaming)
• Mathematicians are creating models to help
explain different electrical rhythms seen in
different stages of sleep
Stages of sleep for humans
• we
focus on the transitions between sleep/wake and REM/NREM
sleep
• sleep patterns of rodents are similar, with more brief awakenings
POAH
Model (Kumar, Bose, Mallick 2012)
MRF
ORX
HOM
MRF
ORX
POAH
CRF
CIRC
CRF
GABA
R-ON
R-OFF
GABASNr
inhibition
excitation
Wakefulness associated areas
: Mid brain reticular formation
: Lateral hypothalamic
orexinergic neurons.
Sleep associated areas
: Preoptic anterior hypothalamus
: Caudal reticular formation
REM sleep associated areas
R-ON
: LDT/ PPT cholinergic neurons
R-OFF
: LC Noradrenergic neurons
GABA
: GABAergic inter neurons in LC
Other inputs
Hom
: Homeostatic sleep drive
Circ
: Circadian clock from SCN
GABASNr : GABaergic neurons from
substantia niagra pars
reticulata
Feed-forward control of one flip-flop circuit by another
Sleep/Wake Transitions
REM-OFF
POAH
MRF
REM-ON
sleep
wake
Fast transitions controlled by POAH
Consistent with Szymusiak et al 1998
wake
sleep
No REM activity in this trace
Transitions to REM sleep
wake
GABA-ergic input from SNR can instigate REM-on activity consistent with
the Pal & Mallick (2009) conjecture
Sleep deprivation
Prolonging the wake state leads to a longer subsequent sleep episode
See Phillips & Robinson (2008) for a systematic study of effects of sleep deprivation
Loss of orexin input to sleep promoting areas
disrupts sleep/wake transitions
gopi = 1.0
gopi = 0.6 gopi = 0.1
Consistent with studies of narcolepsy (Peyron et al 2000)`
Neuroscience to graph theory
• Gap-junctions are physical connections between neurons that allow
current to flow between them
Cell 1
Cell 2
• Can an action potential in cell 1 evoke an action potential from cell 2?
• Dynamics of gap-junctionally connected neurons have been subject of
prior investigation (Sherman-Rinzel 93, Chow-Kopell 98, Lewis et al.
2000, 01, 03, Medvedev et al. 2000, 08, 10, and many more)
• Lewis and Rinzel (2000) asked the question of whether periodic
activity could be sustained in a network of neurons connected by gapjunctions. For specific network architectures (like cycles) they provide
estimates on frequency based on rates of spontaneous activity.
• Gansert, Nadim and Golowasch (2007) asked how the size and shape
of a neuron affects the ability of these networks to sustain activity.
Motivation
Figures from Gansert et. al. (2007)
“kernel” of sustained activity
Some interesting questions in the context
of generating rhythmic activity
●
●
●
●
Are there specific architectures that promote sustained
rhythmic activity? (Cycles for example)
How do rules of nodal interaction affect the global
dynamics ?
How important are intrinsic dynamics of individual
node in the sustainment of activity?
In what way are dynamics related to graph structure?
Graph Properties
• G(n,p)={all graphs with n nodes and a probability
p of an edge between any two nodes}
• The graph property Q consists of a subset of
G(n,p) that share a common feature.
Ex. Q1 = {all graphs with a triangle}
Q2 = {all graphs that are connected}
G1
G1,G 4  Q2
G2
G3
G2,G4  Q1
G4
G3  Q 1, Q2
Thresholds for 3 important monotone properties
(Erdos, Renyi 1960)
• A property Q is monotone if whenever G  Q and G
 H, then H  Q.
• A function p*(n) is said to be a threshold function for
a monotone property Q if
p(n)/p*(n) 0 implies that almost no G
has Q, and
p(n)/p*(n)  implies that almost every
G has Q
• Appearance of the first edge at p ~ O(1/n2)
• Appearance of k-cycles at p ~ O(1/n)
• Disappearance of last isolated node at p ~ O((log
n)/n)
Relating Dynamics and Graph structure
• In Singh et al (2011, SIADS), we show how
random graph structure is related to periodic
activity for both spiking and bursting neurons.
• Very non-intuitive results arise involving the
giant component of the random graph
• We are currently investigating several
theoretical questions in this area.
• Network and graph theory has seen a lot of
interdisciplinary work in the area of physics.
Conclusion
• Mathematics turns out to be a good language
to understand neuroscience.
• Mathematical modeling in close conjunction
with experimental work is beginning to make
inroads into the understanding of biological
systems. There is still a lot of work to be done.
• Science can be advance by considering
interdisciplinary approaches.