BME 6938 Neurodynamics Instructor: Dr Sachin S Talathi

Neuronal communication  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:  Electrical Synapses  Chemical Synapses

Cartoon of synaptic transmission

Electrical synapse  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:

Chemical Synapse  Complex process that begins with the action potential arriving at the pre-synaptic terminal.  The basic story can be summarized in the following steps:  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  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  g syn is the maximal synaptic conductance, V syn 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  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 T max is the maximum amount of transmitters that can be released into the cleft, V p and K p determine the stiffness and the threshold for release. Typical values are V p =2 mV, K p =5 mV and T max =1 mM. Note: Another functional form for T used in modeling is

What current will appear in the post- synaptic cell  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  GABA A receptor activated through gamma aminobutyric acid  GABA B receptor activated through gamma aminobutyric acid

AMPA synapses  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  Popular to model through the alpha-functions  Another version of the synapse model will be discussed in the class

NMDA synapses  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

GABAergic Synapses  Major class of inhibitory synapses in the CNS  GABA A Fast inhibition with kinetics similar to that for AMPA synapse  Reversal potential is close to the reversal potential of potassium channel, V syn =-80 mV  Acts to inhibit the post-synaptic target (not always true)  GABA synapse can be modeled as

GABA-B synapses  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

Synaptic Depression  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 The slower the, the longer the synapse remains in the desensitized state: x

Let’s play with different synapse models in XPPAUTO  Load the file ModelSynapse.ode and try to answer the following questions:  Run simulations for individual active synapses with g=  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  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 12 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 =10 We will see few examples of how complex the dynamics of this 10 dimensional nonlinear system can get gsyn1 gsyn2

Set the system up in XPPAUTO for the following set of ODE’s  Neuron Model  Synapse Model Analyze above system in XPPAUT. Download the file TwoCellNetwork.ode and try to answer the following questions

Lets play with the model  Change initial condition V1 from -67 to -60. Run the network. Use Graph add curve to add neuron 2 voltage to the trace.  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?  Now set gsyn1=gsyn2=0.1 and Integrate again. What happens?  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?  Now set i1=i2=0.5, alpha1=alpha2=3, beta1=beta2=1, and gsyn1=gsyn2=.1? What happens?  Change beta1=beta2=0.1. Integrate again! Now what happens?  Now set alpha1=alpha2=1, beta1=beta2=.2, gsyn1=gsyn2=0.05 and i1=1 and i2=1.05. Integrate again! Now what happens?  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?  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 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  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 (  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.