
NeuroSimulatorA biological neural network simulator based on the HodgkinHuxley modelPiotr Mirowski, New York University, 2006The following computing project was done for the course "Mathematical Aspects of Neurophysiology", taught by Prof. Charles Peskin at the Courant Institute of Mathematical Sciences. At any given neuron i, those equations are: Synaptic transmission between any neurons i and j was modeled as a change in synaptic conductivity G_synapse[i,j](t), trigered by a decreasing change of voltage (at the peak and tail of an action potential). To do so, we input the derivative of the voltage of neuron i into a negative sigmoid function. However, because of the steep peak of the action potential followed by a fast return to rest voltage, the resulting synaptic conductance would have a steep peak as well, and we want that peak to be gentle enough (otherwise, an action potential could be trigerred at the following neuron). In order to smooth the G_synapse[i,j](t) function, we further control its evolution by a firstorder differential equation, according to the following equations: At a given time step t, and after trial and error, we propose to update the value of G_synapse[i,j](t) for each couple of neurons (i,j) according to the following empirical equations: We notice that all the synapses originating from the same neuron have the same conductance at a given time step t. Moreover, we define 2 possible values for the reversal potential E_[j,i](t): a value of 73mV for inhibitory synapses (the IPSP has an amplitude of 5mV) and a value of 60mV for excitatory synapses (the EPSP has an amplitude of 8mV). References: Project code (in Matlab):The entire code is contained in a zip file: NeuroSimulator.zip.
Simulation results2neuron modelsExplanation of the architecture: Experience01_Architecture.pdf. Response of the stimulated neuron to a current clamp: time plot and (s,v) plane plot. Response of the stimulated neuron to a current increase (by a factor 100): time plot and (s,v) plane plot. So far, those experiments show that the rate of firing increases with the intensity of the current stimulation, within an interval (below a certain intensity, the neuron does not fire, and above another intensity, the amplitude of the AP decreases until the neuron does not fire anymore and is clamped at a voltage higher than the rest potential). Response of the stimulated neuron to a random (stochastic) current shocks of same intensity: time plot and (s,v) plane plot, and response of the second neuron (connected to the first neuron by an excitatory synapse): time plot and (s,v) plane plot. 7neuron modelExplanation of the architecture: Experience10_Architecture.pdf. Response of the stimulated neuron to a current shock: time plot neuron 1, and propagation of the action potential in the subsequent neurons of the chain: time plot neuron 2, time plot neuron 3, time plot neuron 4, time plot neuron 5, time plot neuron 6, time plot neuron 7. 4neuron model with 1 excitatory, 1 inhibitory neuronsExplanation of the architecture: Experience11_Architecture.pdf. Neurons 1 (excitatory) and 2 (inhibitory) are stimulated by a shock, and their response, alongside those of neurons 3 and 4 is visualized: 4neuron model with 2 excitatory, 1 inhibitory neuronsExplanation of the architecture: Experience12_Architecture.pdf. Neurons 1 and 3 (excitatory) and 2 (inhibitory) are stimulated by a shock, and their response, alongside those of neurons 4 is visualized: 9neuron model with 7 excitatory, 2 inhibitory neuronsExplanation of the "chessboard" architecture: Experience13_Architecture.pdf. Neuron 1 (excitatory) is stimulated by a shock, and the propagation of its action potential to the excitatory or inhibitory neurons of the 3x3 network is visualized: 49neuron model with 44 excitatory, 5 inhibitory neuronsExplanation of the "chessboard" architecture: Experience14_Architecture.pdf. Neuron 1 (excitatory) is stimulated by a shock, and the propagation of its action potential to the excitatory or inhibitory neurons of the 9x9 network is visualized. As we can see on this map showing the number of times each neuron has fired, the action potential of neuron 1 (lowermost leftmost) manages to propagate until neuron 49 (uppermost rightmost) on the grid Experience14_Firings.pdf: 