- Email: [email protected]
Social barriers to a theoretical neuroscience
TINS August 2000
20/7/00
11:36 am
Page 334
S OUNDINGS Social barriers to a theoretical neuroscience Gin McCollum Social rather than scientific barriers are impeding neuroscience theory. There are plenty of experimental data and mathematical methods to develop a rigorous, mathematical theory in neuroscience. However, structural mathematical efforts are being suffocated by the requirement to produce numbers immediately. Also theoretical development is tied too closely to one experimental group.The social barriers can be addressed by: (1) judging theory by structural accuracy rather than numerical output; (2) recognizing mathematical theory (not just computational modeling) as a method for producing insight into neurobiological phenomena; (3) funding fundamental theoretical neuroscience and (4) recognizing theoretical neuroscientists as neuroscientists. Trends Neurosci. (2000) 23, 334–336
‘R
Gin McCollum is at the Neurological Sciences Institute, Oregon Health Sciences University, Portland, OR 97209, USA.
334
EALISTIC’ has become confused with ‘quantita tive’. Brief examination reveals that the ‘theoretical neuroscience’ featured in many meeting announcements, job applications and reviews, usually refers only to computational modeling. Mathematical theory is flourishing in neuroscience only in those areas where it can be confused with computation. For example, in the current version of Hodgkin–Huxley type modeling, beautiful dynamic mathematics characterize the physiology of excitable cells. However, this is only because the phenomena can be structured as simple resistance–capacitance (RC) electrical circuits, lend themselves to the protocols of electrophysiology and can be precisely quantified. The quantitatively murky area between cell physiology and behavior requires theory based on mathematics that matches the phenomena structurally, before computations are appropriate. Mathematical approaches developed to address issues in sensorimotor neuroscience, by theorists immersed in neuroscience, are difficult to fund or to publish in neuroscience journals. A typical example is the rhythm-space method for predicting rhythms supported by neural circuits: although the first modeling paper was published in a neuroscience journal1, the mathematical treatment was not. Published in a physics journal2, it is out of the view of most neuroscientists. Even perfectly standard, but not necessarily quantitative mathematical methods, such as group theory, are viewed with suspicion by many neuroscientists. Group theory clarifies the structural aspects of the various quadruped gaits3–6. The primary distinctions among gaits, such as walk, trot, and pace, are structural (i.e. in the ordering of legs); quantitative methods to distinguish gaits are secondary. If theoretical neuroscientists using mathematics with structural accuracy could make a living, the group theory of gait would be a small industry, developing the mathematics that connects disparate neurobio logical phenomena. For example, gait would be connected group-theoretically with the detailed physiology of spinal interneurons. Critics ask that the experimental predictions inherent in published papers be explicit. However, experimental tests are the province of experimenters, not theorists. The best way to judge theory is by how accurately the mathematical structure matches neural phenomena.
TINS Vol. 23, No. 8, 2000
Theoretical neuroscience expresses biological insights mathematically Once the organizational structure of a phenomenon is known, mathematics can solidify that knowledge into a domain of inference. Schwartz7 summarizes the advantage of using mathematical language to accurately represent the important elements of a system and the relationships between them: ‘1. Because of mathematics’ precise, formal character, mathematical arguments remain sound even if they are long and complex. In contrast, common sense arguments can generally be trusted only if they remain short… 2. The precisely defined formalisms of mathematics discipline mathematical reasoning, and thus stake out an area within which patterns of reasoning have a reproducible, objective character… 3. Though founded on a very small set of fundamental formal principles, mathematics allows an immense variety of intellectual structures to be elaborated. Mathematics can be regarded as a grammar defining a language in which a very wide range of themes can be expressed. Thus mathematics provides a disciplined mechanism for devising frameworks into which the facts provided by empirical science will fit and within which originally disparate facts will buttress one another…’
Neurobiological facts, framed mathematically, lead to structural biological insights. Besides the Hodgkin–Huxley model, structural insights include Newton’s force law: (F = ma). The proportionality constant (mass; m) is independent of the composition of the object, giving the relationship between force (F) and acceleration (a) a structure beyond the details of the specific example. A more recent example is chaos theory, called at an earlier stage ‘qualitative dynamics’8. When quantitative methods were developed, they were at first abused, leading to confusion between chaos and noise9. By knowing structure, present uses include control of cardiac chaos10.
Structurally accurate theory reliably produces testable predictions Like experimental research, theoretical research follows known procedures to unknown results. The procedures can be summarized in four steps (Fig. 1): (1) problem identification; (2) exposure to phenomena;
0166-2236/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved.
PII: S0166-2236(00)01616-7
TINS August 2000
20/7/00
11:36 am
Page 335
G. McCollum – Theoretical neuroscience
Theory is suffocated by quantitative standards The development of theory requires all four steps for continual improvement of structural accuracy. Short-loop modeling, in which only one metaphor or mathematical formalism is used (Fig. 1), is essential for finding the extent to which a model matches one type of phenomenon. For example, linear systems techniques were at first applied to the oculomotor system as a creative act12, but have become quantitatively routine. Quantitative similarity does not imply structural similarity. Computers have inspired many with a sense that it is possible to understand nervous systems, but they have no necessary structural analogy to nervous system function. Structural accuracy is an especially important requirement in adaptive and learning systems, such as nervous systems, because they vary quantitatively, allowing numerical matching without structural accuracy. Structural mathematical research can also be short-loop, for example when an experimental group hires a mathematician to refine a particular model. Like computational research, such constrained theoretical research is important in the field. The danger is that only these types of theory will be funded and published. Fundamental theoretical research seldom produces numbers immediately. Nor does it typically belong to a predictable loop between experiment and model refinement. Requiring quantitative outputs and shortloop data-model relationships prematurely puts the
Experience Observation Experimental data
r t lo op
Primary concepts Problem identification
Preliminary modeling
Sho
(3) preliminary modeling and (4) model refinement. Volterra followed similar procedures in founding the theory of population dynamics11. (1) Problem identification involves finding a point of view from which order is evident in a domain, so that organizational principles can be expressed mathematically. Illuminating a phenomenon from a new point of view combines conceptual and mathematical reframing. Reframing the problem and constructing the mathematical framework require a set of skills complementary to those of experimental research. (2) Further, more intensive exposure to the phenomena is necessary after problem identification. Although the problem could not be identified without previous exposure to the phenomena, tentative reframing suggests further questions, making this phase particularly fruitful for a theoretical scientist to become intimately familiar with the system under study. (3) Preliminary modeling involves the initial choice of a framework, including mathematical formalism (such as group theory or dynamics) or a metaphor (such as electrical circuits or digital computers). Preliminary modeling is not the application of a procedure or algorithm, but is a creative act, like the choice of words to describe a scene or the choice of an experimental design to investigate a phenomenon. A preliminary model or framework is chosen to express the organization of the system under investigation and to integrate a range of phenomena, giving insight beyond the data under study. (4) Model refinement revises the model or framework to increase the explanatory range and the precision of the conceptual paradigm. Because preliminary modeling reaches into boundary areas beyond the data being modelled, other data become relevant as the model is developed.
SOUNDINGS
Model refinement
Integration with other domains of inference trends in Neurosciences
Fig. 1. Development of mathematical frameworks. Each box represents a step in the process of developing a mathematical framework, starting with primary concepts drawn from empirical observations. Broken lines indicate short-loop computation and modeling.
cart before the horse, repels theorists from the field and prevents theoretical neuroscience from being published and funded.
The responsibility of theoretical neuroscientists is to increase biological insight Successful theory requires the daring to address the phenomena and the perseverance to capture them in mathematics. Experimental neuroscience has illuminated a wealth of intricate organization; theoretical neuroscience needs to model that organization in detail, using mathematics with structural accuracy. This modeling provides challenges, not scientific barriers. Failure to solve a problem from one point of view, using one kind of mathematics, does not mean the problem is intractable. Only by taking the long view has physics attained quantitative precision. In quantum field theory, it is not the obvious quantities that are computed, such as the mass or charge of the electron. Rather, by developing structural theory, physicists arrived at methods for identifying and quantifying the magnetic moment of the electron, leading to the most accurate quantitative prediction by theoretical science13. Quantitative confirmation followed a long structural development. A truly integrated neuroscience will connect the many specialized domains of neurobiological knowledge, and will ultimately be one in which we recognize ourselves through all the intricacies of anatomy and physiology. In cognitive neural networks, so many levels of analysis already exist that lack of engagement with biology goes unnoticed. This leads to a sense and fear that ‘theory’ means only armchairing or vitalism. (These are two conceptual traditions peculiar to neuroscience, although formerly found in all sciences. They start with ideas, often teleological ones, and search for evidence, rather than anchoring concepts directly in neural mechanisms.) Theoretical neuroscience could be so much more: a liberating force encouraging all neuroscientists to respect bio logical insight and to take more time for unification, integration and depth. TINS Vol. 23, No. 8, 2000
335
TINS August 2000
SOUNDINGS
20/7/00
11:36 am
Page 336
G. McCollum – Theoretical neuroscience
Mathematical theory unifies Unified theory grows from mathematical formalizations of separate domains. Mathematics unifies by distilling the logical structure of the organization in each domain, which encourages deeper understanding. From the resulting deeper points of view overlaps between disparate fields of inquiry or between competing models become evident14. A skilled theorist recognizes tractable problems that have the promise of leading to a deeper understanding and unification. To allow unifying theory, neuroscience needs to nurture a rich mathematical context of many structural models. Einstein’s work arose in a rich mathematical context, with intellectual forebears such as Maxwell, Lorenz and Poincaré. Maxwell combined the electrical and magnetic laws into electromagnetic theory, a theoretical cantilever to predict the structure of the photon (and thence the speed of light). Einstein stepped outside electromagnetic theory and Newtonian mechanics to find a point of view from which they are unified, as they are in nature15. Neuroscience offers ample complexity and unity for such theoretical feats. However, theoretical neuroscientists typically operate in a mathematically impoverished context. Part of the difficulty is the misapprehension that there can be only one correct view of a neurobiological system. It is the mark of a complex system to require multiple views16,17. The goal is not one all-encompassing theory, certainly not at first. Rather, rigorous mathematical efforts gradually gather and coalesce.
A dearth of theorists using structural mathematics To produce rigorous mathematical theory, theoretical neuroscience would benefit from an infusion of young mathematicians and theoretical physicists with expertise in sensitively applying mathematics to empirical science. They need to be apprenticed for three or four years to experienced theoretical neuroscientists using mathematics with structural accuracy (which exist) who can encourage them wisely to develop their own paradigms. Theorists beginning independent research should not be forced to follow well-trodden approaches. Programs in which mathematicians, theoretical physicists and others work with experimental groups, engineers or computationalists for one or two years are much too short and too timid to produce rigorous mathematical theory. To create neuroscience theory, it is not the knowledge of physical theories and how to use them that is needed, but the ability to create theory.
‘How can I build one?’ is different from ‘How does it work?’
Acknowledgements The manuscript has benefited from critique by Patrick D. Roberts, Curtis Bell, Gerhard Magnus, Lee T. Robertson, Neal Barmack, F. Owen Black and William Roberts.
336
Although biomedical science needs many types of people for various functions, engineers do not typically produce mathematical neuroscience theory. For example, engineers use physical calculations in the design of rockets, in lieu of more tests. But computing to see how a nervous system works is different from computing how an as-yet-unbuilt rocket will work. The physical principles of rocket functioning are known. What is not known is whether the engineers have combined the components felicitously, to yield the desired performance. Nervous systems, by contrast, are already known to work; it is the principles that are not known. The TINS Vol. 23, No. 8, 2000
essential mistake in considering theoretical neuroscience as nothing but computation is the assumption that the principles of neuroscience are known. Rather, our main problem is to learn the fundamental principles. For this, neuroscience needs a social ecology of many types of people, including theoretical neuroscientists.
Theoretical neuroscientists are neuroscientists, rather than mathematicians or physicists Just as a theoretical physicist is a physicist, a theoretical neuroscientist is a neuroscientist. The expertise of theoretical neuroscientists as both theorists and neuroscientists needs to be recognized. Neuroscience needs to recognize and fund theoretical neuroscience as part of itself. Although mathematicians and theoretical physicists recognize the value of structural mathematics, that is not enough to allow theoretical neuroscience to grow. Nascent mathematical neuroscience is not always deep enough as mathematics to be published in mathematics or physics journals. Although biophysics, the physics of biological structures, constitutes an overlap area between biology and physics, biophysics must not be confused with theoretical neuroscience. The theory of essentially neurobiological phenomena, such as sensorimotor control or cognition, is not physics but neuroscience. Unlike physics, neuroscience is a frontier on which scientific concepts of the nature of life will develop naturally. Neuroscience needs to nurture this conceptual frontier wisely. There are plenty of data and tractable scientific problems. Removing social impediments will free neuroscience to use mathematical tools to accelerate its conceptual development. Selected references 1 Roberts, P.D. (1997) Classification of rhythmic patterns in the stomatogastric ganglion. Neuroscience 81, 281–296 2 Roberts, P.D. (1998) Rhythmic behavior generated by ensembles of neurons. Int. J. Theor. Phys. 37, 3051–3068 3 Schöner, G. et al. (1990) A synergetic theory of quadrupedal gaits and gait transitions. J. Theor. Biol. 142, 359–391 4 Collins, J.J. and Stewart, I.N. (1992) Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion. J. Math. Biol. 30, 827–838 5 Collins, J.J. and Stewart, I.N. (1993) Coupled nonlinear oscillators and the symmetries of animal gaits. Nonlinear Sci. 3, 349–392 6 Golubitsky, M. et al. (1998) A modular network for legged locomotion. Physica D 115, 56–72 7 Schwartz, J.T. (1986) Economics, mathematical and empirical. In Discrete Thoughts: Essays on Mathematics, Science, and Philosophy (Kac, M. et al., eds), p.117, Birkhäuser 8 Gilmore, R. (1981) Catastrophe Theory for Scientists and Engineers, John Wiley & Sons 9 Lindner, J.F. et al. (1998) Can neurons distinguish chaos from noise? Int. J. Bifurcation Chaos 8, 767–781 10 Garfinkel, A. et al. (1992) Controlling cardiac chaos. Science 257, 1230–1235 11 Millan Gasca, A. (1996) Mathematical theories versus biological facts: a debate on mathematical population dynamics in the 1930s. Hist. Studies Phys. Biol. Sci. 26, 347–403 12 Robinson, D. (1977) Vestibular and optokinetic symbiosis: an example of explaining by modeling. In Control of Gaze by Brainstem Neurons, Developments in Neuroscience (Vol. 1) (Baker, R. and Berthoz, A., eds), pp.49–58, Elsevier 13 Pais, A. (1986) Inward Bound: Of Matter and Forces in the Physical World, Oxford University Press 14 Rota, G-C. (1997) The phenomenology of mathematical proof. In Indiscrete Thoughts (Palombi, F., ed.), Birkhäuser 15 Gardner, H. (1993) Creating Minds, Basic Books 16 Segel, L.A. (1995) Grappling with complexity. Complexity 1, 18–25 17 Auyang, S.Y. (1998) Foundations of Complex-System Theories, Cambridge University Press
20/7/00
11:36 am
Page 334
S OUNDINGS Social barriers to a theoretical neuroscience Gin McCollum Social rather than scientific barriers are impeding neuroscience theory. There are plenty of experimental data and mathematical methods to develop a rigorous, mathematical theory in neuroscience. However, structural mathematical efforts are being suffocated by the requirement to produce numbers immediately. Also theoretical development is tied too closely to one experimental group.The social barriers can be addressed by: (1) judging theory by structural accuracy rather than numerical output; (2) recognizing mathematical theory (not just computational modeling) as a method for producing insight into neurobiological phenomena; (3) funding fundamental theoretical neuroscience and (4) recognizing theoretical neuroscientists as neuroscientists. Trends Neurosci. (2000) 23, 334–336
‘R
Gin McCollum is at the Neurological Sciences Institute, Oregon Health Sciences University, Portland, OR 97209, USA.
334
EALISTIC’ has become confused with ‘quantita tive’. Brief examination reveals that the ‘theoretical neuroscience’ featured in many meeting announcements, job applications and reviews, usually refers only to computational modeling. Mathematical theory is flourishing in neuroscience only in those areas where it can be confused with computation. For example, in the current version of Hodgkin–Huxley type modeling, beautiful dynamic mathematics characterize the physiology of excitable cells. However, this is only because the phenomena can be structured as simple resistance–capacitance (RC) electrical circuits, lend themselves to the protocols of electrophysiology and can be precisely quantified. The quantitatively murky area between cell physiology and behavior requires theory based on mathematics that matches the phenomena structurally, before computations are appropriate. Mathematical approaches developed to address issues in sensorimotor neuroscience, by theorists immersed in neuroscience, are difficult to fund or to publish in neuroscience journals. A typical example is the rhythm-space method for predicting rhythms supported by neural circuits: although the first modeling paper was published in a neuroscience journal1, the mathematical treatment was not. Published in a physics journal2, it is out of the view of most neuroscientists. Even perfectly standard, but not necessarily quantitative mathematical methods, such as group theory, are viewed with suspicion by many neuroscientists. Group theory clarifies the structural aspects of the various quadruped gaits3–6. The primary distinctions among gaits, such as walk, trot, and pace, are structural (i.e. in the ordering of legs); quantitative methods to distinguish gaits are secondary. If theoretical neuroscientists using mathematics with structural accuracy could make a living, the group theory of gait would be a small industry, developing the mathematics that connects disparate neurobio logical phenomena. For example, gait would be connected group-theoretically with the detailed physiology of spinal interneurons. Critics ask that the experimental predictions inherent in published papers be explicit. However, experimental tests are the province of experimenters, not theorists. The best way to judge theory is by how accurately the mathematical structure matches neural phenomena.
TINS Vol. 23, No. 8, 2000
Theoretical neuroscience expresses biological insights mathematically Once the organizational structure of a phenomenon is known, mathematics can solidify that knowledge into a domain of inference. Schwartz7 summarizes the advantage of using mathematical language to accurately represent the important elements of a system and the relationships between them: ‘1. Because of mathematics’ precise, formal character, mathematical arguments remain sound even if they are long and complex. In contrast, common sense arguments can generally be trusted only if they remain short… 2. The precisely defined formalisms of mathematics discipline mathematical reasoning, and thus stake out an area within which patterns of reasoning have a reproducible, objective character… 3. Though founded on a very small set of fundamental formal principles, mathematics allows an immense variety of intellectual structures to be elaborated. Mathematics can be regarded as a grammar defining a language in which a very wide range of themes can be expressed. Thus mathematics provides a disciplined mechanism for devising frameworks into which the facts provided by empirical science will fit and within which originally disparate facts will buttress one another…’
Neurobiological facts, framed mathematically, lead to structural biological insights. Besides the Hodgkin–Huxley model, structural insights include Newton’s force law: (F = ma). The proportionality constant (mass; m) is independent of the composition of the object, giving the relationship between force (F) and acceleration (a) a structure beyond the details of the specific example. A more recent example is chaos theory, called at an earlier stage ‘qualitative dynamics’8. When quantitative methods were developed, they were at first abused, leading to confusion between chaos and noise9. By knowing structure, present uses include control of cardiac chaos10.
Structurally accurate theory reliably produces testable predictions Like experimental research, theoretical research follows known procedures to unknown results. The procedures can be summarized in four steps (Fig. 1): (1) problem identification; (2) exposure to phenomena;
0166-2236/00/$ – see front matter © 2000 Elsevier Science Ltd. All rights reserved.
PII: S0166-2236(00)01616-7
TINS August 2000
20/7/00
11:36 am
Page 335
G. McCollum – Theoretical neuroscience
Theory is suffocated by quantitative standards The development of theory requires all four steps for continual improvement of structural accuracy. Short-loop modeling, in which only one metaphor or mathematical formalism is used (Fig. 1), is essential for finding the extent to which a model matches one type of phenomenon. For example, linear systems techniques were at first applied to the oculomotor system as a creative act12, but have become quantitatively routine. Quantitative similarity does not imply structural similarity. Computers have inspired many with a sense that it is possible to understand nervous systems, but they have no necessary structural analogy to nervous system function. Structural accuracy is an especially important requirement in adaptive and learning systems, such as nervous systems, because they vary quantitatively, allowing numerical matching without structural accuracy. Structural mathematical research can also be short-loop, for example when an experimental group hires a mathematician to refine a particular model. Like computational research, such constrained theoretical research is important in the field. The danger is that only these types of theory will be funded and published. Fundamental theoretical research seldom produces numbers immediately. Nor does it typically belong to a predictable loop between experiment and model refinement. Requiring quantitative outputs and shortloop data-model relationships prematurely puts the
Experience Observation Experimental data
r t lo op
Primary concepts Problem identification
Preliminary modeling
Sho
(3) preliminary modeling and (4) model refinement. Volterra followed similar procedures in founding the theory of population dynamics11. (1) Problem identification involves finding a point of view from which order is evident in a domain, so that organizational principles can be expressed mathematically. Illuminating a phenomenon from a new point of view combines conceptual and mathematical reframing. Reframing the problem and constructing the mathematical framework require a set of skills complementary to those of experimental research. (2) Further, more intensive exposure to the phenomena is necessary after problem identification. Although the problem could not be identified without previous exposure to the phenomena, tentative reframing suggests further questions, making this phase particularly fruitful for a theoretical scientist to become intimately familiar with the system under study. (3) Preliminary modeling involves the initial choice of a framework, including mathematical formalism (such as group theory or dynamics) or a metaphor (such as electrical circuits or digital computers). Preliminary modeling is not the application of a procedure or algorithm, but is a creative act, like the choice of words to describe a scene or the choice of an experimental design to investigate a phenomenon. A preliminary model or framework is chosen to express the organization of the system under investigation and to integrate a range of phenomena, giving insight beyond the data under study. (4) Model refinement revises the model or framework to increase the explanatory range and the precision of the conceptual paradigm. Because preliminary modeling reaches into boundary areas beyond the data being modelled, other data become relevant as the model is developed.
SOUNDINGS
Model refinement
Integration with other domains of inference trends in Neurosciences
Fig. 1. Development of mathematical frameworks. Each box represents a step in the process of developing a mathematical framework, starting with primary concepts drawn from empirical observations. Broken lines indicate short-loop computation and modeling.
cart before the horse, repels theorists from the field and prevents theoretical neuroscience from being published and funded.
The responsibility of theoretical neuroscientists is to increase biological insight Successful theory requires the daring to address the phenomena and the perseverance to capture them in mathematics. Experimental neuroscience has illuminated a wealth of intricate organization; theoretical neuroscience needs to model that organization in detail, using mathematics with structural accuracy. This modeling provides challenges, not scientific barriers. Failure to solve a problem from one point of view, using one kind of mathematics, does not mean the problem is intractable. Only by taking the long view has physics attained quantitative precision. In quantum field theory, it is not the obvious quantities that are computed, such as the mass or charge of the electron. Rather, by developing structural theory, physicists arrived at methods for identifying and quantifying the magnetic moment of the electron, leading to the most accurate quantitative prediction by theoretical science13. Quantitative confirmation followed a long structural development. A truly integrated neuroscience will connect the many specialized domains of neurobiological knowledge, and will ultimately be one in which we recognize ourselves through all the intricacies of anatomy and physiology. In cognitive neural networks, so many levels of analysis already exist that lack of engagement with biology goes unnoticed. This leads to a sense and fear that ‘theory’ means only armchairing or vitalism. (These are two conceptual traditions peculiar to neuroscience, although formerly found in all sciences. They start with ideas, often teleological ones, and search for evidence, rather than anchoring concepts directly in neural mechanisms.) Theoretical neuroscience could be so much more: a liberating force encouraging all neuroscientists to respect bio logical insight and to take more time for unification, integration and depth. TINS Vol. 23, No. 8, 2000
335
TINS August 2000
SOUNDINGS
20/7/00
11:36 am
Page 336
G. McCollum – Theoretical neuroscience
Mathematical theory unifies Unified theory grows from mathematical formalizations of separate domains. Mathematics unifies by distilling the logical structure of the organization in each domain, which encourages deeper understanding. From the resulting deeper points of view overlaps between disparate fields of inquiry or between competing models become evident14. A skilled theorist recognizes tractable problems that have the promise of leading to a deeper understanding and unification. To allow unifying theory, neuroscience needs to nurture a rich mathematical context of many structural models. Einstein’s work arose in a rich mathematical context, with intellectual forebears such as Maxwell, Lorenz and Poincaré. Maxwell combined the electrical and magnetic laws into electromagnetic theory, a theoretical cantilever to predict the structure of the photon (and thence the speed of light). Einstein stepped outside electromagnetic theory and Newtonian mechanics to find a point of view from which they are unified, as they are in nature15. Neuroscience offers ample complexity and unity for such theoretical feats. However, theoretical neuroscientists typically operate in a mathematically impoverished context. Part of the difficulty is the misapprehension that there can be only one correct view of a neurobiological system. It is the mark of a complex system to require multiple views16,17. The goal is not one all-encompassing theory, certainly not at first. Rather, rigorous mathematical efforts gradually gather and coalesce.
A dearth of theorists using structural mathematics To produce rigorous mathematical theory, theoretical neuroscience would benefit from an infusion of young mathematicians and theoretical physicists with expertise in sensitively applying mathematics to empirical science. They need to be apprenticed for three or four years to experienced theoretical neuroscientists using mathematics with structural accuracy (which exist) who can encourage them wisely to develop their own paradigms. Theorists beginning independent research should not be forced to follow well-trodden approaches. Programs in which mathematicians, theoretical physicists and others work with experimental groups, engineers or computationalists for one or two years are much too short and too timid to produce rigorous mathematical theory. To create neuroscience theory, it is not the knowledge of physical theories and how to use them that is needed, but the ability to create theory.
‘How can I build one?’ is different from ‘How does it work?’
Acknowledgements The manuscript has benefited from critique by Patrick D. Roberts, Curtis Bell, Gerhard Magnus, Lee T. Robertson, Neal Barmack, F. Owen Black and William Roberts.
336
Although biomedical science needs many types of people for various functions, engineers do not typically produce mathematical neuroscience theory. For example, engineers use physical calculations in the design of rockets, in lieu of more tests. But computing to see how a nervous system works is different from computing how an as-yet-unbuilt rocket will work. The physical principles of rocket functioning are known. What is not known is whether the engineers have combined the components felicitously, to yield the desired performance. Nervous systems, by contrast, are already known to work; it is the principles that are not known. The TINS Vol. 23, No. 8, 2000
essential mistake in considering theoretical neuroscience as nothing but computation is the assumption that the principles of neuroscience are known. Rather, our main problem is to learn the fundamental principles. For this, neuroscience needs a social ecology of many types of people, including theoretical neuroscientists.
Theoretical neuroscientists are neuroscientists, rather than mathematicians or physicists Just as a theoretical physicist is a physicist, a theoretical neuroscientist is a neuroscientist. The expertise of theoretical neuroscientists as both theorists and neuroscientists needs to be recognized. Neuroscience needs to recognize and fund theoretical neuroscience as part of itself. Although mathematicians and theoretical physicists recognize the value of structural mathematics, that is not enough to allow theoretical neuroscience to grow. Nascent mathematical neuroscience is not always deep enough as mathematics to be published in mathematics or physics journals. Although biophysics, the physics of biological structures, constitutes an overlap area between biology and physics, biophysics must not be confused with theoretical neuroscience. The theory of essentially neurobiological phenomena, such as sensorimotor control or cognition, is not physics but neuroscience. Unlike physics, neuroscience is a frontier on which scientific concepts of the nature of life will develop naturally. Neuroscience needs to nurture this conceptual frontier wisely. There are plenty of data and tractable scientific problems. Removing social impediments will free neuroscience to use mathematical tools to accelerate its conceptual development. Selected references 1 Roberts, P.D. (1997) Classification of rhythmic patterns in the stomatogastric ganglion. Neuroscience 81, 281–296 2 Roberts, P.D. (1998) Rhythmic behavior generated by ensembles of neurons. Int. J. Theor. Phys. 37, 3051–3068 3 Schöner, G. et al. (1990) A synergetic theory of quadrupedal gaits and gait transitions. J. Theor. Biol. 142, 359–391 4 Collins, J.J. and Stewart, I.N. (1992) Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion. J. Math. Biol. 30, 827–838 5 Collins, J.J. and Stewart, I.N. (1993) Coupled nonlinear oscillators and the symmetries of animal gaits. Nonlinear Sci. 3, 349–392 6 Golubitsky, M. et al. (1998) A modular network for legged locomotion. Physica D 115, 56–72 7 Schwartz, J.T. (1986) Economics, mathematical and empirical. In Discrete Thoughts: Essays on Mathematics, Science, and Philosophy (Kac, M. et al., eds), p.117, Birkhäuser 8 Gilmore, R. (1981) Catastrophe Theory for Scientists and Engineers, John Wiley & Sons 9 Lindner, J.F. et al. (1998) Can neurons distinguish chaos from noise? Int. J. Bifurcation Chaos 8, 767–781 10 Garfinkel, A. et al. (1992) Controlling cardiac chaos. Science 257, 1230–1235 11 Millan Gasca, A. (1996) Mathematical theories versus biological facts: a debate on mathematical population dynamics in the 1930s. Hist. Studies Phys. Biol. Sci. 26, 347–403 12 Robinson, D. (1977) Vestibular and optokinetic symbiosis: an example of explaining by modeling. In Control of Gaze by Brainstem Neurons, Developments in Neuroscience (Vol. 1) (Baker, R. and Berthoz, A., eds), pp.49–58, Elsevier 13 Pais, A. (1986) Inward Bound: Of Matter and Forces in the Physical World, Oxford University Press 14 Rota, G-C. (1997) The phenomenology of mathematical proof. In Indiscrete Thoughts (Palombi, F., ed.), Birkhäuser 15 Gardner, H. (1993) Creating Minds, Basic Books 16 Segel, L.A. (1995) Grappling with complexity. Complexity 1, 18–25 17 Auyang, S.Y. (1998) Foundations of Complex-System Theories, Cambridge University Press