TA: John Olthoff
The goal of this project is to address the following two questions:
· How learning can occur at the cellular level?
· How can this be modeled and simulated quickly using the Izhikevich model?
Quick, successive stimulation of neurons in the hippocampus, a brain region associated with learning and memory, can lead to increases in excitatory synapse strengths which can last between hours and weeks (Malenka). This phenomenon, called long term potentiation (LTP), has been proposed as a cellular explanation for some forms of learning, such as classical conditioning and associative learning.
LTP has been showed to occur at synapses with post-synaptic glutamate receptors called NMDA (N-methyl-D-aspartate) receptors and AMPA (a-amino-3-hydroxy-5-methyl-4-Isoxazolepropionic) receptors (Malenka). Glutamate, released from the pre-synaptic terminal, binds to both NMDA and AMPA receptors on the post-synaptic terminal. When glutamate binds to AMPA receptors, the receptor is activated and Na+ can flow into the cell and K+ can flow out (Xiao). AMPA receptors are responsible for the majority of the inward current.
NMDA and AMPA Receptors:
NMDA receptors are voltage dependent and are only activated by glutamate during a simultaneous post-synaptic potentiation. Normally, an extracellular Mg2+ ion blocks the NMDA channel, but a depolarization can remove the ion, allowing Na+ and some Ca2+ into the dendritic spine and K+ out of the spine (Xiao).
Figure 1: A simple model of NMDA Receptors
(Image from: Robert C. Malenka, et al. “Long-Term Potentiation: A Decade of Progress?” Science. 285, 1870 (1999).)
It is thought that the increase of Ca2+ is the critical factor which initiates LTP. Ca2+ is localized to a single synapse and this explains why repetitive stimulation at a particular synapse increases only the strength of that particular synapse. LTP is associative, which means a strong depolarization at a particular synapse can lead to an increase in strength of another synapse if both are stimulated simultaneously. LTP is cooperative, which means that a single weak stimulus cannot induce LTP (Bliss).
NMDA receptors require a simultaneous pre-synaptic and post-synaptic potentiation to activate and this feature allows NMDA receptors to act like coincidence detectors. The influx of Ca2+ by the opening of the NMDA channel can initiate an enzymatic cascade which eventually leads to the long term increase in the number of AMPA and NMDA receptors. Calcium binds to calmodulin, which activates CaMKII. CaMKII phosphorylates itself, which explains the sustained effect of LTP after Ca2+ levels have decreased. CaMKII also phosphorylates AMPA receptors and is thought to lead to an increase in the number of AMPA receptors at the synapse (Malenka).
Figure 2: A simple model of the signal chain which can lead to LTP (Malenka)
(Image from: Robert C. Malenka, et al. “Long-Term Potentiation: A Decade of Progress?” Science. 285, 1870 (1999).)
The Izhikevich model is a dynamical systems approach to modeling neuronal firing. It simplifies the Hodgkin and Huxley model in order to increase computational speed, while still retaining biological realism. In the model each neuron is represented by two state variables, v and u, which are both dependent on each other and are updated at each time-step. The state variable v can be easily understood as the voltage of the neuron, while u is important for resetting the voltage after a neuron fires.
In classical conditioning, an arbitrary stimulus is paired with a meaningful stimulus and over time, the arbitrary stimulus acquires the properties of the important stimulus. Eventually the arbitrary stimulus can elicit the same response as the important stimulus. The unconditioned stimulus (US) is the stimulus that naturally elicits the innate response called the unconditioned response (UR). The conditioned stimulus (CS),the arbitrary stimulus, eventually elicits the automatic response which is then called the conditioned response (CR).
Probably the most famous example of classical conditioning is Pavlov’s dog. In this experiment Pavlov presented a dog with food, the US, and this elicited salivation, the UR. Next the food was paired with a tone, the CS. After time, the tone presented alone could elicit salivation, the CR.
The first step was to represent the NMDA receptor in the Izhikevich model. Because NMDA receptors cannot be activated when the Mg2+ ion blocks its pore, the conductance of NMDA receptors is voltage dependent. In attempt to make the NMDA receptor biologically realistic, the NMDA receptor conductance in this model is voltage and time dependent, as is true of NMDA receptors in biological systems. In two articles, one by Saudargiene and the other by Zachor, NMDA receptor conductance is represented as:
These values were determined from actual voltage clamp studies in the hippocampus (Saudargiene). To incorporate this equation into the Izhikevich model, it was modified slightly in interest of simplicity and computational speed. The maximum conductance at each voltage value was pre-calculated and only the larger of the time constants is used. The modified equation for conductance incorporated into this model is as follows:
In this model NMDA receptors were made to recognize NMDA and glutamate.
The focus of this project was on the NMDA receptor. In order to keep the code simple, to represent other receptors like the AMPA receptor, a modification of the current-based Izhikevich model was used. At each synapse, the type of neurotransmitter released from the pre-synaptic terminals and the concentration of each type of receptor at the post-synaptic terminal can be arbitrarily defined. An arbitrary amount of receptors can be defined at each synapse and each can recognize an arbitrary amount of neurotransmitters. Also, each receptor can recognize neurotransmitters that make inhibitory and excitatory connections.
In this model, AMPA receptors could recognize AMPA and glutamate.
For simplicity’s sake, Izhikevich’s current-based sense-input model is retained. Inputs from senses can be defined in terms of currents.
The standard Izhikevich model has two state variables, v and u. A state variable is added to represent the calcium concentration in each synapse which is dependent on the NMDA receptors that are active, the conductance of NMDA, the post-synaptic potential, and a decay constant:
In this model
When the concentration of Ca2+ reaches a critical threshold value in a given post-synaptic neuron at a specific synapse, the strength of that synapse is increased. Depending on the simulation, this critical threshold was varied. Values of 0.3 and 0.5 were used in the simulations below. Increases in synaptic strength are represented in the code by increasing the concentration of the AMPA receptors at that synapse. The strength of this synapse is quickly increased when Ca2+ is above the threshold. This increase in strength continues long after the calcium leaves the cell. The strength of the synapse slowly decays to its initial value with a long time-constant.
The synaptic strength is represented as:
Where represents the strength of the AMPA receptors and represents the change in strength. and is the amount increased each ms, scaled by the number of steps per ms. is the decay constant. In this model, was varied depending on the specific simulation. Typical values used were between 50 and 100.
A three neuron classical conditioning learning circuit was implemented as follows:
Figure 3: A simple classical learning circuit.
US represents the unconditioned stimulus, CS represents the conditioned stimulus, and R represents the response which could be either the unconditioned or conditioned response.
The modified formula for NMDA conductance can be written as:
After calculating for each voltage value, both the modified formula for conductance and Saudargiene’s formula were plotted against voltage and time:
Figure 4: Saudargiene’s NMDA Receptor Conductance compared with the Modified NMDA Receptor Conductance
When plotted, it is obvious that the modified equation for NMDA conductance is similar to the equation used by Saudargiene. The difference between the two plots is minimal and this simplified formula was substituted.
Figure 5: Post-Synaptic Voltage, and Pre-Synaptic Voltage plotted against Post-Synaptic Calcium Concentration
When the pre and post-synaptic neurons fire within a finite time frame, calcium enters the post-synaptic NMDA receptors cell. In this model, this finite time frame was chosen as 10 time steps. The pre-synaptic cell must fire before the post-synaptic cell for calcium to enter the cell. This agrees with spike timing dependent plasticity, which states that the pre-synaptic neuron must take part in firing the post-synaptic neuron. The concentration of calcium in the post-synaptic dendrite decreases exponentially over time. The amount of calcium that enters the cell depends on the voltage of the post-synaptic receptor and whether it is above
Figure 6: Voltage vs. Time of a Pre-Synaptic neuron that synapses on two post-synaptic neurons.
The plot above shows the voltage in
3 neurons. The first is the pre-synaptic neuron which synapses on two other
neurons. One of these neurons has NMDA and AMPA at the synapse, while the other
only has AMPA receptors. All three neurons briefly receive sense input, and the
first neuron receives ¼ of the synaptic input from there-on afterwards. In this
brief period, the post-synaptic neuron with NMDA receptors is depolarized by
the sense input and the simultaneous input from the pre-synaptic current sets
up the proper conditions for LTP. During this period, the synapse is briefly
strengthened. Over time, the strength of the post-synaptic NMDA synapse returns
to its initial value and fires at the same rate as the neuron that does not
have NMDA. In contrast, after the brief period of simultaneous stimulation, the
post-synaptic neuron with only AMPA receptors fires at the same rate
throughout, so there is no strengthening of the synapse. The length of the time
that the LTP lasts can be easily manipulated by changing the AMPA receptor
strength decay constant.
Figure 7: Post-Synaptic concentration and post-synaptic AMPA receptor concentration vs. time.
Initially the concentration of calcium increases with time during the period of simultaneous pre and post-synaptic excitation. After that period, the calcium decays exponentially. Once the calcium is above the threshold level, the concentration of AMPA receptors begins to increase in a fashion that resembles the 1- function. The concentration of AMPA receptors continues to increase even as the calcium begins to decay. Eventually the AMPA receptor concentration begins to decrease exponentially.
Self Induced LTP:
Figure 8: Self Induced LTP – LTP can be induced in a neuron with NMDA and AMPA receptors by rapidly stimulating the dendrite. A neuron without NMDA receptors does not show self induced LTP.
In this model, self induced LTP can be demonstrated. Both neurons in the above figure received a strong sense stimulus for a brief period and then a weak stimulus for the remaining period. The neuron that shows LTP has NMDA receptors while the neuron that does not show LTP does not have NMDA receptors. Because of the way this model is coded, the NMDA receptors must be added to a self synapse for this self-induced LTP to work.
Therefore in this model both ways of inducing LTP can be represented. Rapid stimulation of a neuron or stimulating a neuron that is simultaneously depolarized will both induce LTP. Both of these methods can induce LTP in biological systems.
Classical Conditioning Learning Circuit:
In this following example, a Ca2+ threshold of 0.5 and an AMPA receptor strength decay length of 100 was used.
Figure 9: A Simple Classical Conditioning Learning Circuit. This figure shows voltages in all three neurons of the classical conditioning circuit and regions of interest are highlighted.
Using this model, learning at the cellular level can be modeled and simulated quickly within the framework of Izhikevich’s dynamical systems approach. A simple classical learning circuit was successively created by integrating LTP into the Izhikevich model.
There are several ways this model could be improved. This model can only represent LTP and would have to be extended in order to represent LTD. It would be interesting to create a larger learning circuit with more complicated connections and experiment with how the synaptic weights change over time given different inputs. Perhaps the ability to grow new connections could be added. One thing that would have been interesting to experiment with is ocular dominance patterns, which arise through similar mechanisms.
This archive contains the 16 MATLAB files necessary to run the code which will display Figure 9. After downloading and extracting type ‘LTP_sim’ into the MATLAB command window. To plot the curves type ‘calc_g_max’.
Bliss, T.V.P and G.L Collingridge. “A Synaptic Model of Memory: Long Term Potentiation in the Hippocampus.” Nature. 361, (1993).
Robert C. Malenka, et al. “Long-Term Potentiation: A Decade of Progress?” Science. 285, 1870 (1999).
Saudargiene, Ausra, et. al. “Biologically Inspired Artificial Neural Network Algorithm Which Implements Local Learning Rules.” Proceedings of the 2004 International Symposium on Circuits and Systems. 5 (2004) 389-392.
Xiao, Min-Yi. “Comparing fluctuations of synaptic responses mediated via AMPA and NMDA receptor channels—implications for synaptic plasticity.” BioSystems. 62 (2001) 45–56.
Below is a link containing more information about the Izhikevich model, including MATLAB code:
Simple Model of Spiking Neurons: