320 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 4, NO. 5, OCTOBER 2010 Dynamics and Bifurcations in a Silicon Neuron Arindam Basu, Member, IEEE, Csaba Petre, Graduate Student Member, IEEE, and Paul E. Hasler, Senior Member, IEEE Abstract—In this paper, the nonlinear dynamical phenomenon associated with a silicon neuron are described. The neuron has one transient sodium (activating and inactivating) channel and one ac- tivating potassium channel. These channels do not model specific equations; instead they directly mimic the desired voltage clamp responses. This allows us to create silicon structures that are very compact (six transistors and three capacitors) with activation and inactivation parameters being tuned by floating-gate (FG) tran- sistors. Analysis of the bifurcation conditions allow us to identify regimes in the parameter space that are desirable for biasing the circuit. We show a subcritical Hopf-bifurcation which is character- istic of class 2 excitability in Hodgkin-Huxley (H-H) neurons. We also show a Hopf bifurcation at higher values of stimulating cur- rent, a phenomenon also observed in real neurons and termed exci- tation block. The phenomenon of post-inhibitory rebound and fre- quency preference are displayed and intuitive explanations based on the circuit are provided. The compactness and low-power na- ture of the circuit shall allow us to integrate a large number of these neurons on a chip to study complicated network behavior. Index Terms—Bifurcations, ion-channel dynamics, nonlinear modeling, silicon neuron. I. MODELING BIOLOGICAL NEURONS IN SILICON T HIS paper explores the nonlinear dynamics of a silicon neuron that exploits the similarity between the transport of ions in biological channels and charge carriers in transistor channels. While basic results for this circuit, such as voltage clamp responses, were presented in [1], we show the similarity of this circuit with biology and the Hodgkin-Huxley (H-H) model across different parameter regions. Thus, this paper strongly validates the idea of modeling biological channels using transistors by providing a dynamical equivalence be- tween models. Moreover, the differential equations describing the circuit that we develop place it in a dynamical systems framework accessible to computational neuroscientists and mathematicians enabling collaborations with circuit designers. Pioneering work in modeling the dynamics of a biological neuron was performed by Hodgkin and Huxley in the 1950s. Since then, mathematical models of varying complexities have been proposed and studied, facilitated largely by the Manuscript received May 16, 2009. Date of current version September 29, 2010. This paper was recommended by Associate Editor T. Roska. A. Basu is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail:arindam.basu@gmail. com). C. Petre is with the Brain Corp., San Diego, CA 92121 USA (e-mail: cpetre3@gatech.edu). P. Hasler is with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-250 USA (e-mail: phasler@ece.gatech.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBCAS.2010.2051224 rapid growth of digital computers. However, simulating a large number of these “mathematical” neurons is still a daunting task, especially when there are multiple time scales involved in the differential equations. Thus there has also been a trend toward implementing the modeling equations on an analog platform, the earliest instance of which dates back to Richard Fitzhugh in the 1960s. Also with the progress in semiconductor technology, power-supply levels have become similar to biology, opening a new possibility of the same silicon model to interface with live neurons [2], [3] . All of the implementations have rigorously followed the H-H equations or a simplified version of the same. However, we use a silicon neuron that uses a metal–oxide semiconductor field-effect transistor (MOSFET) to represent a channel [1] since carrier transport phenomenon are similar in both. Different control amplifiers are used to control the gate of these “channel” transistors so that the voltage clamp responses of the individual channels resemble biology. A combination of one sodium and one potassium channel makes a basic neuron as shown in Fig. 1. While [1] demonstrated voltage-clamp data similar to biology, we present analytical and experimental bifur- cation diagrams with varying input current that matches biology and the H-H model. Also, the analytical framework developed in this paper allows easy identification of valid biasing regimes of the circuit. Instead of proving the topological equivalence to H-H equations, we concentrate on the bifurcations exhibited by this circuit and use it as a metric to validate the efficacy of this model. An introduction to this approach was presented in [4] where biasing conditions for Hopf bifurcations were explored. This paper presents a complete analysis of the center-manifold reduction, bifurcation diagrams based on continuation, a de- scription of the designed chip and modifications to the circuit for independent parameter control, and measurement results from a silicon prototype. The neuron is also biased in an excitable regime where it displays phenomenon, such as post inhibitory rebound and frequency preference. This approach of studying bifurcations is useful because it is believed that computational properties of neurons are based on the bifurcations exhibited by these dynamical systems in re- sponse to some changing stimulus [5], [6] leading one to be- lieve that all models which present the same set of bifurcations should be equally good in analyzing and modeling neurons. For example, any neuron exhibiting a Hopf bifurcation can easily signal when a stimulus crosses a threshold by initiating a spike train, while those exhibiting saddle-node bifurcations can en- code the strength of a stimulus in their firing rate. Hence, by showing that this silicon neuron has bifurcations similar to a certain class of biological neurons, we can claim that the silicon neuron can also perform similar computations. A similar approach has been employed in [7] but they do not explore the property of excitation block. In fact, we propose an 1932-4545/$26.00 © 2010 IEEE