Special issue on applied neurodynamics: from neural dynamics to neural engineering
Hillel J. Chiel, Peter James Thomas · Journal of Neural Engineering · 2011
Tracing technologies back in time to their scientific and mathematical origins reveals surprising connections between the pure pursuit of knowledge and the opportunities afforded by that pursuit for new and unexpected applications. For example, Einstein's desire to eliminate the disparity between electricity and magnetism in Maxwell's equations impelled him to develop the special theory of relativity (Einstein 1922)Einstein 1922 p 41 'The advance in method arises from the fact that the electric and magnetic fields lose their separate existences through the relativity of motion. A field which appears to be purely an electric field, judged from one system, has also magnetic field components when judged from another inertial system.'. His conviction that there should be no privileged inertial frame of reference Einstein 1922 p 58 'The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered must be considered as small in comparison with this progress.' further impelled him to utilize the non-Euclidean geometry originally developed by Riemann and others as a purely hypothetical alternative to classical geometry as the foundation for the general theory of relativity. Nowadays, anyone who depends on a global positioning system—which now includes many people who own smart phones—uses a system that would not work effectively without incorporating corrections from both special and general relativity (Ashby 2003). As another example, G H Hardy famously proclaimed his conviction that his work on number theory, which he pursued for the sheer love of exploring the beauty of mathematical structures, was unlikely to find any practical applications (Hardy 1940)Hardy 1940 pp 135–6 'The general conclusion, surely, stands out plainly enough. If useful knowledge is, as we agreed provisionally to say, knowledge which is likely, now or in the comparatively near future, to contribute to the material comfort of mankind, so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. Modern geometry and algebra, the theory of numbers, the theory of aggregates and functions, relativity, quantum mechanics—no one of them stands the test much better than another, and there is no real mathematician whose life can be justified on this ground. If this be the test, then Abel, Riemann and Poincaré wasted their lives; their contribution to human comfort was negligible, and the world would have been as happy a place without them.'. Ironically, the famous Rivest, Shamir and Adleman (RSA) algorithm, which currently underpins much of modern cryptography, depends on fundamental ideas from number theory (Cormen et al 2001). Finally, the indeterminacy of the quantum states of light, atoms and molecules, a source of great theoretical interest in the first quarter of the last century, is now in the process of being harnessed for creating algorithms, and novel computers, that can solve problems that could not be addressed by current computing devices (Steane 1998, Ralph and Pryde 2010). Thus, perhaps we should not be surprised that a focus on whether a three-body system (such as the sun, earth and moon) would remain stable over time ultimately became the basis for a new geometrical way of thinking about nonlinear dynamical systems, and that this approach has begun to find practical applications in the understanding and control of nervous systems, including novel ideas for brain–computer interfaces. Classical dynamical systems theory began with the work of Newton on the motion of the planets. He was able to solve a two-body problem, the motion of the earth around the sun (Newton 1687, Chandrasekhar 1995). Finding explicit solutions for the slightly more complicated problem of three bodies (for example, the sun, earth and moon) proved to be far more difficult. In the late nineteenth century, Poincaré made significant progress on this problem, introducing a geometric method of reasoning about solutions to differential equations (Diacu and Holmes 1996). This work had a powerful impact on mathematicians and physicists, and also began to influence biology. In his 1925 book, based on his work starting in 1907, and that of others, Lotka used nonlinear differential equations and concepts from dynamical systems theory to analyze a wide variety of biological problems, including oscillations in the numbers of predators and prey (Lotka 1925). Although little was known in detail about the function of the nervous system, Lotka concluded his book with speculations about consciousness and the implications this might have for creating a mathematical formulation of biological systems. Much experimental work in the 1930s and 1940s focused on the biophysical mechanisms of excitability in neural tissue, and Rashevsky and others continued to apply tools and concepts from nonlinear dynamical systems theory as a means of providing a more general framework for understanding these results (Rashevsky 1960, Landahl and Podolsky 1949). The publication of Hodgkin and Huxley's classic quantitative model of the action potential in 1952 created a new impetus for these studies (Hodgkin and Huxley 1952). In 1955, FitzHugh published an important paper that summarized much of the earlier literature, and used concepts from phase plane analysis such as asymptotic stability, saddle points, separatrices and the role of noise to provide a deeper theoretical and conceptual understanding of threshold phenomena (Fitzhugh 1955, Izhikevich and FitzHugh 2006). The Fitzhugh–Nagumo equations constituted an important two-dimensional simplification of the four-dimensional Hodgkin and Huxley equations, and gave rise to an extensive literature of analysis. Many of the papers in this special issue build on tools directly descended from the analysis of the Hodgkin and Huxley equations in FitzHugh and Nagumo's early work. Mathematicians became increasingly interested in biological problems in general, and in the function of the nervous system in particular, during the latter part of the twentieth century. The natural tool for describing more complex neural systems whose patterns of activity unfold in time was nonlinear dynamical systems theory. Classic work from such investigators as Kolmogorov, Arnol'd, Moser, Malkin, Andronov, Hopf, Birkhoff, Hartman and others (reviewed in Izhikevich 2006) served as the basis for understanding the dynamics of neural models such as the coupling of oscillators for rhythmic behavior, leading to work such as that of Koppell and Ermentrout on the lamprey swimming system (Kopell and Ermentrout 1986, 1990), based on earlier models of Cohen et al (1982). Exploration of nonlinear interactions in neuronal populations, especially those that might be related to vision, led to the development of the Wilson–Cowan equations in the 1970s (Wilson and Cowan 1972, 1973). The advent of increasingly powerful personal computers also made it feasible to combine theoretical analyses with extensive numerical investigations of nonlinear dynamical systems. An important and influential example of such work was the detailed bifurcation analysis of Morris and Lecar's two-dimensional model of nonlinear dynamical behavior in the giant muscle fiber of the Pacific barnacle Balanus nubilis (Morris and Lecar 1981), done by Rinzel and Ermentrout in the late 1980s (Rinzel and Ermentrout 1989). The mathematical analysis of bursting behavior based on decomposition of a dynamical system into fast and slow subsystems, an application of Fenichel's geometric singular perturbation theory (Fenichel 1979, Jones 1995), continues to play an important role. Recent work on dynamical analyses of neurons and neural circuits is described in Izhikevich's recent book (Izhikevich 2006), which is based in part on his own work in this area. This is a very small glimpse of a much larger literature; these mathematical themes recur throughout this issue. Practitioners of neural engineering who want to explore the language and role of dynamics further can find accessible introductions to the key ideas in works such as Strogatz (1994) and Izhikevich (2006). In this special issue of Journal of Neural Engineering , we provide a sample of the vigor and excitement of the recent developments in the applications of nonlinear dynamical systems theory to the understanding and control of the nervous system. Four of the papers demonstrate the power of dynamical systems theory to analyze and understand neural systems, both in isolation and within a neuromechanical context (Coggan et al 2011, Nadim et al 2011, Spardy et al 2011a, 2011b). One paper focuses on the importance of noise and delay in dynamical systems for control (Milton 2011). Two papers focus on the dynamics of ion channels—in one paper, new approaches for estimating their parameters are described (Meng et al 2011), and in a second, the time courses of sodium ion channels are used to understand conduction block due to high-frequency stimulation (Ackermann et al 2011). Two papers focus on the use of optimal control theory to develop approaches for understanding (deWolf and Eliasmith 2011) and controlling (Nabi and Moehlis 2011) the nervous system. Finally, two papers begin to explore longer time scale neural dynamics through a combination of modeling and experiments, examining how animals learn to reduce the time required to forage for food at multiple sites (de Jong et al 2011), and how the dynamics of the respiratory system change with development (Fietkiewicz et al 2011). The first four papers of this special issue illustrate the use of dynamical systems theory to analyze and understand neural circuitry and neuromechanical systems. The first of these papers uses the phase response curve (PRC) of an oscillator, which is a conceptual tool rooted in the analysis of systems of nonlinear differential equations that quantifies the effect of internally generated or externally applied perturbations on the phase of an ongoing oscillation. Nadim et al (2011) elegantly apply PRCs and related techniques to shed light on mechanisms for stabilizing the period of a particular central pattern generator circuit, responsible for the pyloric rhythm in the stomatogastric ganglion of the crab. Although the digestive system of Cancer borealis may seem somewhat removed from the concerns of neural engineers, this system has provided the basis for both experimental and theoretical work on the role of neuromodulation on neural circuitry. Neuromodulators can functionally alter the dynamics of a neural circuit on a moment-to-moment basis, 'carving out' distinct functional circuits from a single anatomical circuit (Marder and Thirumalai 2002). Furthermore, a hallmark of central pattern generator (CPG) systems in humans and other animals is a balance of robustness to perturbation and adaptability to changing conditions. Here Nadim et al (2011) focus on robustness, both to intrinsic perturbations such as barrages of irregular synaptic activity (incorporated into a dynamical systems model of the circuit as a Poisson input train), as well as perturbing inputs from other rhythmic processes internal to the animal, such as a slow modulatory input from the animal's gastric mill. Experimentally identified inhibitory feedback from a particular part of the circuit to a pair of pacemaker cells appears extraneous at first, inasmuch as suppressing it seems to have no effect on the period of the pyloric rhythm. But while the mean period is unaffected by removing this synapse's effect, the variance of the period shows great sensitivity. Using an experimental approach that allows them to artificially remove the a model that is based in a on the system, the use the of the phase response with and without the to the this inhibitory results can be at a conceptual phase plane analysis to that inhibitory synaptic input and the intrinsic of the to out the in phase by results may have implications for the role of in stabilizing nervous systems, as well as neural The paper in this special issue uses dynamical analysis to shed light on the of for in of or et al (2011) provide into in excitability through dynamical analysis of use of analysis to the and of that may work both with a model and a model based on the model find that to depends on dynamical such as , in which a dynamical system has more than one stable for a of parameters this a stable and an stable find that the behavior depends on geometrical of the such as the between stable and in the phase on their that may have an to due to or may be able to an is for the in the that is of may not be as as time of for whether As et al (2011) in excitability can be in and can be on the basis of a small number of complex nonlinear In could be a for this special further in computing there is and practical to models as models more to analyze any mathematical at and ultimately to when is to the basis for a model with the components may be a models the to apply tools from dynamical systems theory to the nonlinear dynamical basis for the The and papers in this special by Spardy et al use dynamical systems theory to understand the dynamics of the theoretical framework of dynamical systems, a central pattern generator circuit is of as a stable , or this of it is natural to the between the central circuit and the system as real neuromechanical systems feedback from the to the central and these interactions may have for rhythm and et al feedback into a neuromechanical central pattern generator the also can provide a to As an example, et al that an from the over a of to those in a in which feedback from the to the was The mechanisms by which behavior can be both and the of are important for Spardy et al analyze control in a neuromechanical model of that these experimental results Using mathematical techniques from dynamical systems such as of the dynamics into fast and slow subsystems, are able to the geometry between of and of analysis how feedback the of for which stable oscillations In distinct mechanisms by which the rhythmic behavior in the and of in one the is by from and in the other from In a paper, this out a further analysis of the control system, them to the issue. As the of of the motion not change in of motion is by the phase while the phase is it that a of related related rhythm neurons and and is able to a model that the dynamical of the larger are able to a or are not to in the of the as as As part of the and an important the model are able to that the to a more model has not the of and that a in the The paper in this special by the classical dynamical systems approach by incorporating the of and unexpected in the noise and an engineering can within control systems. For example, in a feedback control can be the of in a neural control system is it mathematical analysis of the system. The of and perturbations further system analysis. (2011) gives an of this with applications to human neural control problems, for example the and response time during in of an oscillations and As (2011) in to from analysis of control systems with human it is to understand the nature of the that the nervous system uses to The paper in this special by et al focuses on the problem of for models of cells that have multiple The of the dynamics of which are far more than from the many ion channels within their which them to be to be or based on to in rhythmic and to have their dynamical by In many about the of is can be from these et al (2011) use the theory of processes and to the parameters for a dynamical model of a that it is to two and for a model from two of with two and in which this approach could be to more complex The paper, by et al also focuses on the dynamics of ion channels to shed new light on a the by which high-frequency stimulation can block conduction of action in block techniques for the of by activity from to central Using the known dynamics of sodium and and a analysis of a dynamical conduction the are able to the by which to from into the nervous system. The paper, by and Moehlis uses approaches from control theory to explore the control or of In many a dynamical For example, and with are to activity in of neurons in such as the within the and et al of stimulation is in that it can the system in a small number of can one effectively a dynamical system an input that is to a much the of the system for any and of the neurons and Moehlis (2011) a these by optimal control for a system that of the three oscillators with a to is, a stable and a control that can be applied to one of the is the optimal control a control another with how light a is it to the oscillators from their by a The depends in part on the by the coupling between the For the coupling between the and the others is then from one can that no control the cells are for control the find that control of can effectively the for a of The paper one of the within neural the application of control techniques to dynamics of circuits in the central nervous system. of systems through and the of interactions intrinsic to central neural circuits this problem especially The paper by and Eliasmith (2011) a general theoretical framework for modeling neural the neural optimal control on concepts from optimal control theory. Using the key control concepts for for a and in which this could be neural systems, and apply their framework to understand both in and in those by and also that of the nonlinear of neurons during behavior are a natural of the control problem by the neural as have Although much experimental work to their the that ideas from control theory could be into novel interfaces. Finally, the last two papers begin to two important of time scale dynamics in the nervous and The paper by Jong et al (2011) a new for understanding how animals to problems by the that food the that animals that are able to find more of in their The problem is the animals to use to the nature of the and then to how to their over If a food source is animals are able to their to to the the food As the this is related to a classic problem in theory, the problem , and may shed light on how biological systems not solutions to such problems, as well as creating a new for understanding the of problems within such as the The paper by et al (2011) uses both and modeling studies to understand the dynamics of the respiratory system. neurons the respiratory system change early in and play an important role in stable Using the demonstrate the of development at which these and then a model that into how this change may be papers also important for work. As (2011) noise and are in the nervous system, the and the and all have to many of the control of the system not be these processes are and et al 2011, 2011, et al the two papers by Spardy et al that nervous systems are and that neural dynamics is by the neuromechanical of the The paper by and Eliasmith (2011) an that optimal control theory and may provide into the nervous system. to be whether this approach can the nature of biological nervous systems, and the complex and dynamics of The papers by Jong et al (2011) and et al (2011) that understanding more about and development provide into the slow dynamics of the nervous system that it to over the of an the of to the global dynamics of the system. Many of the papers and novel to begin to develop new brain–computer based on an understanding of neural parameters for the dynamics the approaches by et al an understanding of the dynamics of ion channels to develop new for or in the nervous system, the approaches by et al an understanding of the dynamics of neural to find the that could or the approach of and Moehlis the importance of to a neural circuit in response to the approach of Nadim et al and for bursting in based on the dynamical analysis of et al Although it may the approaches by these papers that brain–computer not between activity and input or of the by the dynamics of the nervous system. and and analysis of high-frequency conduction block Neural in the global positioning system Chandrasekhar for the H and The has a behavior from interactions of nervous system, and H and The in control and in a context G and A in excitability through dynamical analysis of models Neural Cohen A Holmes and H The nature of the coupling between oscillators of the lamprey generator for a mathematical model and to Jong G and The into the dynamics of in the Neural and Eliasmith The neural optimal control for control Neural and Holmes the of and Einstein A 1922 The of singular perturbation theory for differential equations A and G in change in the respiratory Neural FitzHugh models of threshold phenomena in the and differential models for ion noise in neurons Hardy G H 1940 A with a by and A of neuronal by stimulation Neural Hodgkin A and Huxley A 1952 A quantitative of current and application to conduction and in Izhikevich in The of and Izhikevich and FitzHugh model Jones singular perturbation theory in pp and Ermentrout G and in of oscillators and Ermentrout G and other phenomena in of oscillators Landahl H and Podolsky the of conduction in with Lotka A 1925 of and and Thirumalai synaptic and of neuromodulation Neural A and A control of from a model A and A approach to biophysical neural models from Neural G The and nervous implications for neural control Neural Morris and Lecar H oscillations in the barnacle giant muscle fiber A and Moehlis input optimal control for Neural Nadim and A feedback in an Neural Newton Ralph and Pryde G in pp Rashevsky of pp Rinzel and Ermentrout G of neuronal excitability and oscillations in and pp and H and of the Spardy A and A dynamical systems analysis of control in a neuromechanical model of Neural Spardy A and A dynamical systems analysis of control in a neuromechanical model of Neural A computing Strogatz H and with to and Engineering A for the first time of the process and A noise in neurons H and Cowan and inhibitory interactions in of model neurons H and Cowan A mathematical theory of the functional dynamics of and nervous