Transcript Dynamics of neuronal networks

Introduction to the mathematical
modeling of neuronal networks
Amitabha Bose
Jawaharlal Nehru University &
New Jersey Institute of Technology
[email protected]
IISER, Pune February 2010
Typical Neuron
Applied
Mathematician’s
Neuron
Mathematician’s
Neuron
x'  f ( x)
x  Rn
Why are action potentials important?
•
•
Action potentials are measurable events
The timings or firing rate of action potentials
can encode information
- place cells in hippocampus
- coincidence detection for sound localization
- orientation selectivity in visual cortex
•
Neurons can communicate with one another
using action potentials
Synaptic Communication
Pre-synaptic neuron
Post-synaptic neuron
• An action potential in the pre-synaptic neuron provides a
current to the post-synaptic cell.
• The effect may be either excitatory (naively thought to
promote firing of post-synaptic neuron) or inhibitory
(naively having the opposite effect).
• Synapses have their own time scales for rise and decay.
• Synaptic delays may be involved.
• Synapses can change strength as a function of usage;
this is called synaptic plasticity.
Crustacean Pyloric Rhythm (CPG)
PD
LP
PY
Hooper 94, 95
Bean, Nature Rev. Neuro. 2007
Modeling a neuron as an RC circuit
• Membrane separates charge
• Ions flow through channels
causing voltage changes
I c  Im  Ii
Q  CmV , Ic  dQ / dt
Ii  V / Rm, Rm  1 / g
Cm
dV
 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 )
p
q
INa  g Na mNa (v )hNa (v)[v  VNa ]
IK  g K mK r (v)hK s (v)[v  VK ]
IL  gL[v  VL ]
Outline
• Examples of neuronal computation
• Complications for modeling
• Understanding the Hodgkin-Huxley
equations
• Simple models to analyze synchronous
and anti-phase oscillations
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
Inhibition
Excitation
P = synchronous group of place cells
Place cells
• Place cells also code for location in 2-dimensional
environment. They interact with head direction cells.
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
Visual Cortex
• Neurons fire at preferential orientations
Visual Cortex
• Very well suited to detecting orientations,
contrasts, directions of movement, yet
cannot resolve certain visual scenes.
Necker Cube
Basal Ganglia - Parkinsonian Tremor
• Normal state: Irregular, no correlations in STN cells
• Parkinsonian state: Rhythmic, STN cells cluster
Rubin & Terman, 2004
Central Pattern Generator
• Important for coordinated movement of muscle groups.
• Rhythmic and very stable behavior
Crustacean Pyloric Rhythm (CPG)
PD
LP
PY
Nadim et al,
2000’s
Ventilatory Rhythms
Wilson et al 2002
Buccal Episode
Lung Episode
Lung Area
Buccal Area
Overlaid traces show
that the lung bursts begin
at same point in buccal cycle
as in PIR.
Overlaid traces show that the
buccal cycles continue
predictably from last lung burst.
Different types of bursting neurons
(students - ask Pranay about this!)
Modelers Goldmine or Minefield?
(title borrowed from K.B. Sinha)
•
Many complications: high dimensionality,
multiple time scales, stochasticity, noise, large
networks, unknown architecture…
• Many advantages: “young field”, no canonical
equations thus much freedom, growing number
of interactions with experimentalists
Q. What is the appropriate level of detail for
modeling?
Q. Is there any fun mathematics to be done?
General Biological Questions
• What accounts for the rhythmic activity?
• Under what circumstances do synaptic
and intrinsic properties of neurons
cooperate or compete?
• What effect do multiple time scales have?
• What are the underlying neural
mechanisms that govern behavior?
Translation to mathematics
• What accounts for the rhythmic activity?
Periodic solutions in phase space
• Under what circumstances do synaptic and intrinsic
properties of neurons cooperate or compete?
Effect of parameters on solutions
• What effect do multiple time scales have?
Singular perturbation theory
• What are the underlying neural mechanisms that govern
behavior?
Deriving mathematically minimal models that reveal
necessary and sufficient conditions
To the blackboard!