Opamp circuits:

Repressilator: The repressilator is a genetic oscillator (3). Here we build an electronic approximation of the repressilator. Gene expression is represented by the output of opamps U2, U3 and U5 below. Protein concentration is represented by the output of U1, U4 and U6. If all of the gene representing opamps have absolute gain greater than 2 (e.g. -R2/R1) then the system oscillates. The RC circuits (e.g. R3C1) represent the time constant of production/degradation of protein. LTspice model, which requires the constant module and the noise module explained below. In a real circuit the noise module is not necessary because real resistors are noisy, and so the oscillator starts up quickly. In the simulation, the oscillator can enter a metastable state for a long time until numerical error starts the oscillation.





Filters:

Neural models:

See also a hardware neurons page I generated in 2005.

FHN: Maeda and Makino (1) show how to model a neuron using 3 transistors for a FitzHugh-Nagumo (FHN) type neuron (simplified from Hodgkin-Huxley formulation). The FitzHugh-Nagumo scheme replaces the fast Na current of the HH model with a simplified fast, depolarizing, activation process, and replaces the slow Na inactivation and slow, repolarizing, K current by a single slow inactivation process. The circuit is show below for the FHN neuron. The circuit produces a constant train of simulated action potentials (AP) when a leak current is applied via R4. Note that the amplitude of the simulated action potential is much larger than that of a physiological neuron. Simulated AP amplitude is around 5 volts, while in real life the AP amplitude is around 100 mV. LTspice model.





Adding inactivation: A modification of Maeda and Makino (1) circuit simulates sodium inactvation. Below, the cyan trace is sodium current and the red trace potassium current. LTspice model.





Bursting: By adding one more repolarizing process, modeled by two more transistors, Maeda and Makino (1) can produce a neuron with bursting behavior. LTspice model.





Heart cell: Maeda, Yagi, and Makino (2) extend the model to include heart cells. They slightly modified one of the repolarizing processes to make a plateau potential, similar to that found in a ventricle cell. The implementation below includes a pulse generator which turns on at 4.0 seconds to pace the cell at a rate higher than its endogenous rate. It takes about 3 beats to completely phase-lock. LTspice model.





Simulated ECG: Combining several heart cells into a 2D network (ventricle) and driving them from a simulated atrial pacemaker via a purkinje fiber system, then adding all of the cell potentials and passing them through a lumped model of the body results in a simulated ECG. LTspice model zip file.





Modules:

Computational: The modules in this ZIP file make up a minimal set to converrt LTspice into an analog computer. Unzip them into one directory. The modules define an adder, multiplier, constant, absolute value, and integrator. The damped_osc_test.asc file (in the zip file) can be used to test the modules. Plot the position output versus time, as below. The reset input on the integrators restarts the integrals at the initial conditions, therefore restarting the simulation. The damped spring-mass oscillator equation oscillates if the damping is low enough. To set the value of the constant module to 10 (for example), right-click and enter constant=10 . The average amplitude calculation uses a lowpass filter to detect when to restart the integrators.



