We (Misha Rabinovich, Pablo Varona, Ramon Huerta, Attila Szucs, Rob Elson, and Allen Selverston) have been studying the dynamics of small assemblies of neurons known as Central Pattern Generators (CPGs) which have the job of translating sensory inputs into nearly rhythmic output to muscles. The particular CPG we work with appears in the California spiny lobster Panrulis interruptis.It is called the pyloric CPG, and it has the job of dilating and contracting the pyloric chamber of the lobster to transport shredded food from the stomach to the digestive system.
The pyloric CPG appears in the lobster stomatogastric ganglion and consists of fourteen neurons whose
connections among each other are fully known.
We have investigated the oscillations of these neurons individually as well as in subcircuits of the whole CPG. Our experiments typically take long, clean time series of the voltage across the membrane of the neuron, and for individual neurons typically is chaotic. The detailed modeling study for one of these neurons, the lateral pyloric or LP neuron, using Hodgkin Huxley conductance based models is found in the paper Calcium Dependent Dynamics (PDF Format, Postscript Format) which will soon appear in Biological Cybernetics. In this paper we show that the chaotic oscillations of the LP neuron require some additional slow dynamical process in addition to the usual ion channel dynamics of the Hodgkin Huxley models. We attribute this to a slow exchange of Calcium between the endoplasmic reticulum and the intracellular medium. Further in this paper we show that even though the model has thirteen dynamical variables, it, as does the experiment, shows only a low dimensional attractor of dimension about three to five. This means we can capture the essential membrane voltage behavior of these neurons with a much reduced description.
We have also studied the synchronization properties of these neurons both in their natural state and as we change the sign and magnitude of the couplings between them. In the following paper (Synchronous Behavior of Two Coupled Biological Neurons; Physical Review Letters, Volume 81 pp 5692-5695 (1998)) we analyzed the synchronization of two pyloric dilator or PD neurons from the pyloric CPG which are connected by an electrical or gap junction coupling in the natural state. We added to the natural connection of the neurons which has a conductance of about 200 nS a device which could change the magnitude and sign of this electrical (or gap junction) coupling. When the coupling is strong and positive, the two neurons synchronize their spiking as well as bursting firing patterns. When the coupling is natural, the bursts (slow oscillations) synchronize, while when the coupling is strong and negative, the neurons fire out of phase and regularize their oscillations!
We have performed other such experiments exemplified by the following picture:
On the basis of these experiments and the analysis of many sets of time series of these neurons using the tools described in Analysis of Observed Chaotic Data published by Springer, we have built low dimensional analog electrical versions of these neurons. We call them e-neurons, and their design is based on the work of Hindmarsh and Rose who approximated the current voltage characteristics of these neurons as they appear in Hodgkin Huxley equations by polynomials in the dynamical variables. This is an excellent numerical approximation over the experimental dynamical range for these neurons. Here is an example of the "membrane voltage" versus
for an e-neuron as we vary the injected DC current from positive at the top to negative at the bottom. The e-neurons show the same behavior as individuals as well as in coupled pairs or other subcircuits we have studied in the laboratory. In particular, the bifurcation sequence of two PD neurons in the paper above is captured by two electrically coupled e-neurons.
We have succeeded in coupling the e-neurons to the biological neurons, and even in replacing an e-neuron in the experiments just discussed. For example, here is what happens when an e-neuron (in blue) oscillating chaotically
is coupled to an isolated PD neuron (green trace) which is firing in an irregular, but non-bursting fashion. When the two neurons are coupled (middle of the picture), the PD resumes its bursting/spiking activity, characteristic of its behavior when the neurons are in their natural state.
The first paper on this subject has been submitted for publication. It is here as a PDF file Interacting biological and electronic neurons generate realistic oscillatory rhythms which was published in Neuroreport on February 28, 2000.
Here are some pictures of the electronic neurons. The first is the box with adjustable parameters. Inside the box is a PC board with the analog circuitry.
The next photo is of four 3 dimensional ENs in the experimental configuration:
We have performed a set of experiments which involve the coupling of a pair four dimensional ENs to each other with electrical (ohmic) coupling and with biologically realistic couplings which are either excitatory (cause action potentials at the receiving neuron) or inhibitory (depress action potentials at the receiving neuron). All three of these have been realized in analog circuitry, and the experiments we performed are reported in the PDF file of our paper Synchronous Behavior of Two Coupled Electronic Neurons.
