Transcript BME 6938 Neurodynamics

BME 6938
Neurodynamics
Instructor: Dr Sachin S Talathi
Neuronal communication
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Synapses are principle sites for communication between
neurons (Existing notion that synapses are the basic unit
of computation in the brain…read the interesting article I
put on the course-website: BasicUnitofComputation.pdf)
Types of synapses:
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Electrical Synapses
Chemical Synapses
Cartoon of synaptic transmission
Electrical synapse
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Electrical transmission between two neurons occur at gap junction
between dendritic processes
Gap junction is a membrane protein (connexin) that couples the two
neurons together.
Transmission through electrical synapses is sign conserving
It is suitable for high speed transfer of signals and plays an important role in
synchronizing activity of coupled neurons.
Many non-neural cells (neurons in developing brain) are coupled through
electrical synapses (eg. Glial cells, Muscle cells)
We model electrical synapse by simply adding an additional current to the
current balance equation of a neuron given as:

post
Ielec
 Gelec Vpre  Vpost

Chemical Synapse
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Complex process that begins with the action potential arriving
at the pre-synaptic terminal.
The basic story can be summarized in the following steps:
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AP arriving at the pre-synaptic terminal opens up a no.of calcium ion
channels.
calcium activates a calcium binding protein which promotes release by
binding to vesicles containing the transmitter
these ``docked'' vesicles release their transmitter into the synaptic cleft.
These vesicles bind to their postsynaptic targets (receptors) resulting in
opening of ion channels, and flux of ions into or out of the cell
Our goal is to describe this complicated process
mathematically through a dynamical model
Modeling a Chemical Synapse
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Simple dynamical model that can capture the essence of the processes
described in last slide.
We use a crude approach (that can be considered as a mean field or
average response of a synapse to an action potential) (More details on
various synapse models can be found in a chapter by Destexhe, Mainen, and
Sejnowski posted on the course website: ModelingChemicalSynapses.pdf)
We treat synapse just as another ionic channel that is modeled through
Ohms Law. For every synaptic junction we add the following current to the
current-balance eq. of the post-synaptic neuron

post
Isyn
 gsyn s(t) Vpost  Vsyn
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gsyn is the maximal synaptic conductance,Vsyn is the reversal potential for
the synapse and s(t) is a nonnegative variable, that represents the fraction
of open receptors with bound neurotransmitter

Amount of transmitter released by a
single AP
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A good approximation to the amount of transmitter
released into the synaptic cleft due to a single AP arriving
at the pre-synaptic terminal is given through
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T Vpre
Tmax
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(Vpre  Vp )
1  exp 

Kp

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Tmax is the maximum amount of transmitters that can be released into the
cleft,Vp and
 Kp determine the stiffness and the threshold for release. Typical
values are Vp=2 mV, Kp=5 mV and Tmax=1 mM.
Note: Another functional form for T used in modeling is
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T(Vpre )  0.5 1  tanh 120(Vpre  0.1)
What current will appear in the postsynaptic cell
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Dependent on the type of neurotransmitter released and
the form of receptor on the postsynaptic terminal we we
will consider the following 4 different types of synaptic
responses
AMPA receptor activated through glutamate
NMDA receptor activated through glutamate
GABAA receptor activated through gamma aminobutyric
acid
GABAB receptor activated through gamma aminobutyric
acid
AMPA synapses
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Glutamate is the receptor for AMPA synapse
AMPA currents typically activate and deactivate fast and
the reversal potential for AMPA synapse is around 0 mV.
AMPA currents can be modeled through

post
IAMPA
 gAMPA s(t) Vpost  VAMPA
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ds
 T(1 s)  s
dt
Popular to model through the alpha-functions
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 Another version of the synapse model will be discussed
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in the class
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NMDA synapses
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These are slower excitatory synapses
Play an important role in memory and plasticity (Hebb
Hypothesis)
The difference from AMPA receptors is that the conductance
of NMDA channel depend on post-synaptic membrane
potential in a complex manner through the levels of
magnesium ions in the external medium
I
post
NMDA
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 gNMDA B(Vpost )s(t) Vpost  VNMA
ds
 T(1 s)  s
dt

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BV  
1
e 0.062V Mg
1
3.57
2

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GABAergic Synapses
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Major class of inhibitory synapses in the CNS
GABAA Fast inhibition with kinetics similar to that for
AMPA synapse
Reversal potential is close to the reversal potential of
potassium channel,Vsyn=-80 mV
Acts to inhibit the post-synaptic target (not always true)
GABA synapse can be modeled as
post
GABAA
I
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 gGABAA s(t) Vpost  VGABAA
ds
 T(1 s)  s
dt
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GABA-B synapses
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GBABA-B synaptic transmission involves second
messengers
GABA binds to G-proteins on the post synaptic terminal
that inturn bind to potassium ion channels to open them
up
We can model GABA-B synapse as follows
post
IGABAB
 gGABAB
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sn
Vpost  VGABAA
n
s  Kd
dr
 T(1 r)   ' r
dt
ds
 K 3r  K 4 s
dt
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Synaptic Depression
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Many cortical neurons have AMPA synapses that depress:
(think adaptation), given repeated stimuli the neuron
produces less and less neurotransmitter
This effect can readily modeled by adding a desensitized
state to the standard two-state model for the synapse
 T 
C 
O

O 
X

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2
X 
C
ds
  T 1 s  x  s
dt
dx
 s  2 x
dt
The slower the 2 , the longer the synapse remains in the
desensitizedstate: x
Let’s play with different synapse models
in XPPAUTO
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Load the file ModelSynapse.ode and try to answer the
following questions:
Run simulations for individual active synapses with
g=0.038.
Why is the depolarization with AMPA greater than GABAA?
Look at response to NMDA in zero Mg and GABAB. Why is the response
so small? Why is GABA-B response almost non-existent?
Set ip=35 to generate burst of AP from presynaptic neuron. Compare the
AMPA response to the depressed AMPA response. What is the difference?
Set time scale to 1000 ms and redo the above burst experiment with
NMDA and GABAB synapses. Why is GABAB so much higher with bursts?
No change Mg=1 and look at the response to NMDA.
Synapses in networks
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One of the key questions over last decade or so in
Computational Neuroscience is the behavior of networks of
neurons that oscillate and whether or not can they
synchronize?
One of the interesting theoretical questions is how the
synaptic properties alter the state of synchrony in the
network?
We will look at a very simple network of two cells coupled
through synapse to get an appreciation for the fact that how
complex even most simplest networks can get.
We will introduce some tools that enable us to simplify the
network and enable us to better understand some of the
computational properties of the network later on in the
course
Simple network of mutually coupled
neurons
gsyn1
1
2
gsyn2
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Each neuron A and B is a HH type neuron with 4 dimensional ODE
Synapse from A to B is modeled through a first order kinetic equation
as discussed in last class with parameters gsyn1, Vsyn1, alpha1, beta1
Parameters for synapse from B to A are gsyn2, Vsyn2,alpha2, beta2
Total dimension for the coupled system is 4+4+1+1=10
We will see few examples of how complex the dynamics of this 10 dimensional
nonlinear system can get
Set the system up in XPPAUTO for the
following set of ODE’s
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Neuron Model
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Synapse Model
Analyze above system in XPPAUT. Download the file TwoCellNetwork.ode
and try to answer the following questions
Lets play with the model
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Change initial condition V1 from -67 to -60. Run the network. Use Graph add curve to add
neuron 2 voltage to the trace.
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Now couple neuron 2 to neuron 1 by selecting gsyn1=0.05. What happens? Decrease gsyn1
until neuron 2 no longer fires. You should see a small depolarization of neuron 2 (called EPSP,
in neuroscience world). What is the minimum gsyn1 value to eliminate neuron 2 spike?
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Now set gsyn1=gsyn2=0.1 and Integrate again. What happens?
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Increase gsyn1=gsyn2=0.15 and integrate again. What happens? Are two neurons
synchronized? If so what is the phase difference between firing times of the cells? What
happens if both synapses are inhibitory?
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Now set i1=i2=0.5, alpha1=alpha2=3, beta1=beta2=1, and gsyn1=gsyn2=.1? What happens?
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Change beta1=beta2=0.1. Integrate again! Now what happens?
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Now set alpha1=alpha2=1, beta1=beta2=.2, gsyn1=gsyn2=0.05 and i1=1 and i2=1.05.
Integrate again! Now what happens?
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Now make cell 2 inhibitory by changing vsyn2=-80. Integrate and calculate the period of
oscillations. Slowdown the decay rate of inhibitory synapse by changing beta2=0.1. Recalculate
the period? Why does the period get longer?
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Final example to see how complex things can get. Set Vsyn1=0, Vsyn2=-80, alpha2=0.5,
beta2=0.01, gsyn1=.01, gsyn2=1, i1=3 and i2=0. Set total integration time to 1000. Now
integrate. What do you see? Explain the behavior? How do you modify gsyn1 to get fewer cell
1 spikes for each cell 2 spike?
Even such simple network is complex
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Imagine biological network of billions of neurons with
trillions of synapses!!!
Simulating such a large network and making sense of it
soon becomes exponentially difficult not withstanding
how much computationally expensive simulation of such
network can be … Ambitious goal the Blue Brain Project
(http://bluebrain.epfl.ch/)
Need for tools that simplify the network dynamics.
We will spend some time after the spring break looking
at some mathematical techniques that allow us to analyze
such complex networks in simplified manner.