Hybrid Complex Systems – A Case Study Using Neuronal Dynamics

Nature is full of examples of complex phenomena, which are often studied with the aid of computational models. Consider a model of the spread of an infectious disease among the world's populations. The model specifies such factors as the lifestyles and behaviors of different populations, their geographic locations of the populations, and the modes of interaction between and among populations. Assuming that the model's assumptions are valid, using such models offers many advantages: complete observability of all model variables at all places at all times (how many people in Chicago have the disease 5 days after an infection outbreak), control over the simulation's initial conditions (what if the outbreak began in Seattle instead?), and control over the simulation's parameters (what effect on the spread of the disease would canceling all commercial airline flights have?). Mathematical and computational techniques can be applied to the model to develop a picture of the dynamics of the entire simulated system to determine when bifurcations occur. A bifurcation can be generally described as a set of parameters where a model undergoes a profound change in its dynamics. For example, in the disease model above, let x describe the frequency of interaction between infected and uninfected members of the population. If x is less than a critical value x*, the disease may fade away on its own or be locally contained; but if x is greater than x*, the disease starts spreading at an accelerating rate. In this context, x is a bifurcation parameter and x* is a bifurcation.

Such models have their limits, however. They are only as good as the data that is put into them and the assumptions made by the model creator -- we have all experienced the weather forecast that never came true. To make models more accurate, we need to obtain more data and limit our assumptions; to make them better understood, we often need to
make the models simpler, which may require us to expand our assumptions.

This proposal is concerned with pursuing an alternative approach to understanding complex systems, which we call the hybrid simulation-experiment (HSE). The general idea is to construct a model system that is part experiment and part simulation running in "real-time'' -- the model performs its calculations at the same pace as the experimental elements are changing in time. This approach has been used in various contexts in widely disparate fields, but has not been considered to be a distinct approach to studying complex systems in its own right.

For example, economists have studied the effect of pricing systems on market dynamics -- in some cultures you walk into a store and see a fixed price on the shelf, deciding whether or not to purchase a product. In other cultures negotiating between the buyer and seller is the expected norm. Scientists have studied these modes of buyer/seller interaction using computer simulations interacting with human subjects. The experimental component is the human subjects, who are free to interact with the simulation and make their own choices. The simulation component serves to simulated different pricing systems and control the interaction among subjects. Such a hybrid approaches has many advantages -- by using human subjects many assumptions have been eliminated, with the remaining assumptions confined to the simulation that controls the interactions among users. The simulation component of the HSE approach allows the introduction of controllability and observability (of the simulation components) to an otherwise experimental system.

This approach is limited to particular experimental systems. For example, we could not construct a computer simulation that interacted in any significant way with the climate of North America! What is required is an experimental system whose size and speed is compatibility with technologies that allow interaction with the system and computational technologies that allow simulations to run in real-time at a sufficient speed.

Application

My particular field of research, computational neuroscience, is concerned with understanding the biophysical basis of electrical activity in neurons and how neurons use these electrical properties to communicate with other neurons and process information. Neurons have a voltage across their cell membrane, which can be considered to be a capacitor with several complex nonlinear resistances (referred to by neuroscientists using the reciprocal conductance) in parallel. The neuron may be at rest, which means the voltage is constant. The neuron can also fire an action potential, in which some of the nonlinear resistances rapidly turn on and off, causing a spike-like voltage waveform several milliseconds long. The neuron may spike only in response to input, or it may oscillate, generating a constant repetition of action potentials. We record this voltage by inserting an electrode into the neuron, and we stimulate the neuron by injecting current into the neuron. Neurons can also stimulate each other through synapses, where an action potential in one neuron causes a conductance to change in another neuron.

The general biophysical principals underlying these changed in electrical are well understood, thanks in part to the pioneering work of Hodgkin, Huxley, and Eccles, who shared the Nobel prize in 1963 for their research into the nature of electrical excitability in neurons and interneuronal communication. Starting with these principals, many neuroscientists use computational models to understand the neural systems that they study. Much progress has been made with this approach, but there are limits to what is experimentally testable using conventional techniques.

For example, there are many mathematical and computational models of how neurons generate an action potential. In these models, we typically manipulate one of the aforementioned conductances (g), since that is what synapses do. When g is below a critical value g*, the voltage in the cell is constant. When g is increased greater than g*, the cell oscillates and periodically generates action potentials. Thus at g*, a bifurcation from steady-state silence to oscillatory activity occurs. By the late 1980's, years of modeling and the analysis of such models have revealed two common general mathematical mechanisms for this bifurcation -- a saddle-node bifurcation and a Hopf bifurcation. These bifurcations can be differentiated by such features as the range of frequencies of periodic activity that the cell demonstrates and how the cell response to input when it is electrically silent. But do these two mechanisms exist in nature?

This is where the HSE approach is useful. We can record the voltage from a neuron, read it into a computer, solve a simulation that models one or more of the aforementioned nonlinear conductances, and inject a current into the neuron that is equivalent to the current flowing through one of these simulated conductances. We have artificially added a conductance to the cell we are recording from. We have precise control over this simulated conductance, thus we can systematically vary g and study the resulting electrical properties of the hybrid simulation-neuron, creating a detailed bifurcation diagram of the HSE. Thus the HSE approach can be used to modify the electrical activity of a single neuron, artifically adding conductances or modifying existing conductances within the neuron.

We can also use the HSE to study methods of interneuronal communication. Artificial networks can be created using neurons which are isolated from each other, using the simulation to create artificial connections between neurons. Thus we can have precise control over the strength and speed (kinetics) of simulated synapses between two real neurons. This approach can be used to study under what conditions neurons synchronize their oscillatory activity what form of synchrony occurs: in-phase, where the neurons are spiking at the same time, or anti-phase, where the neurons alternate spiking in a constant back-and-forth relationship.

In the context of this proposal, a final step in this proposal is to attempt to define a science and technology of HSE. How should results be interpreted? The experimental element of the system, the neurons, have a certain intrinsic amount of variability, and there is also measurement error. The simulation portion of the system must be capable of running and responding in real-time, which poses computational challenges. The numerical techniques used to solve the models within the simulation have numerical inaccuracies, and these inaccuracies may be exacerbated by using simpler numerical techniques that make it possible to run the simulation in real-time. Techniques must be developed to quantify the uncertainties, variabilities, and inaccuracies that may be present in the experimental and simulation portions of a HSE.

In conclusion, this project attempts to solve some basic question in computational neuroscience, while developing methodologies that may pave the way to other applications of HSEs to studying complex systems. This approach to investigating complex systems poses many challenges but is an exciting idea that is still in its infancy.