J. J. Hopfield

Neural computation of decisions in optimization problems

Highly-interconnected networks of nonlinear analog neurons are shown to be extremely effective in computing. The networks can rapidly provide a collectively-computed solution (a digital output) to a problem on the basis of analog input information. The problems to be solved must be formulated in terms of desired optima, often subject to constraints. The general principles involved in constructing networks to solve specific problems are discussed. Results of computer simulations of a network designed to solve a difficult but well-defined optimization problem-the Traveling-Salesman Problem-are presented and used to illustrate the computational power of the networks. Good solutions to this problem are collectively computed within an elapsed time of only a few neural time constants. The effectiveness of the computation involves both the nonlinear analog response of the neurons and the large connectivity among them. Dedicated networks of biological or microelectronic neurons could provide the computational capabilities described for a wide class of problems having combinatorial complexity. The power and speed naturally displayed by such collective networks may contribute to the effectiveness of biological information processing.

Transient kinetics of chemical reactions with bounded diffusion perpendicular to the reaction coordinate: Intramolecular processes with slow conformational changes

Intramolecular reactions inside macromolecules (e.g., binding of ligands to iron inside heme proteins) may often be coupled to slow random fluctuations in the reaction center geometry. This motion is ‘‘perpendicular’’ to the reaction coordinate. It can be described as bounded diffusion in the presence of a binding potential field and an intramolecular rate constant which depends on the perpendicular degree of freedom. The diffusion equation is solved under the appropriate reflective boundary conditions. The transient decay of the total population is multiexponential (power law) for small diffusivity, changing to monoexponential kinetics for large diffusivity. For large times or large diffusivity, direct integration is very tedious, but an eigenvalue expansion converges rapidly. It also allows the calculation of the ‘‘average survival time’’ (an extension of the ‘‘first passage time’’) a natural candidate for replacing the reciprocal rate constant in multiexponential kinetics. An example is given for electron transfer between two loosely bound sites in a macromolecule. The average survival time shows a non‐Kramers dependence on diffusivity, of the type found in the binding kinetics in heme proteins.

CO binding to heme proteins: A model for barrier height distributions and slow conformational changes

A model for the dependence of the potential energy barrier on a ‘‘protein coordinate’’ is constructed. It is based on a two dimensional potential energy surface having as variables the CO–iron distance and a conceptual protein coordinate. The distribution of barrier heights observed in kinetics follows from an initial Boltzmann distribution for the protein coordinate. The experimental nonexponential rebinding kinetics at low temperatures or large viscosities (when the protein coordinates can be assumed ‘‘frozen’’) can be fit with a simply parametrized energy surface. Using the same energy surfaces and the theory of bounded diffusion perpendicular to the reaction coordinate, we generate (in qualitative agreement with experiment) the survival probability curves for larger diffusivity, when the constraint on the protein coordinate is relaxed. On the basis of our results, the outcomes of new experiments which examine the concepts underlying the theory can be predicted.

Electron tunneling through covalent and noncovalent pathways in proteins

A model is presented for electron tunneling in proteins which allows the donor–acceptor interaction to be mediated by the covalent bonds between amino acids and noncovalent contacts between amino acid chains. The important tunneling pathways are predicted to include mostly bonded groups with less favorable nonbonded interactions being important when the through bond pathway is prohibitively long. In some cases, vibrational motion of nonbonded groups along the tunneling pathway strongly inluences the temperature dependence of the rate. Quantitative estimates for the sizes of these noncovalent interactions are made and their role in protein mediated electron transport is discussed.

Modeling the olfactory bulb and its neural oscillatory processings

The olfactory bulb of mammals aids in the discrimination of odors. A mathematical model based on the bulbar anatomy and electrophysiology is described. Simulations of the highly non-linear model produce a 35–60 Hz modulated activity which is coherent across the bulb. The decision states (for the odor information) in this system can be thought of as stable cycles, rather than point stable states typical of simpler neuro-computing models. Analysis shows that a group of coupled non-linear oscillators are responsible for the oscillatory activities. The output oscillation pattern of the bulb is determined by the odor input. The model provides a framework in which to understand the transform between odor input and the bulbar output to olfactory cortex. There is significant correspondence between the model behavior and observed electrophysiology.

Artificial neural networks

Examines the following questions associated with artificial neural networks: why people are interested in artificial neural networks; what artificial neural networks are, from the point of view of electronic circuits, and how they work; how they can be programmed and made to solve particular problems; and whether interesting problems can actually be put on such networks. The author then describes the current state of artificial neural network technology and the resulting challenges to people working on electronic devices.<<ETX>>

A theory of edge-emission phenomena in CdS, ZnS and ZnO

Abstract Tentative symmetry assignments of p valence bands and s conduction bands can be made for ZnO and CdS on the basis of a tight-binding model. The six-fold degenerate p -bands are split in hexagonal crystals into a four-fold degenerate and a two-fold degenerate band. The fourfold degeneracy is split by spin-orbit coupling. The polarization of recombination radiation depends upon which band the hole belongs to, and is almost independent of the recombination mechanism. The polarization of the edge emission (the series of equally spaced emission lines) should be strongly temperature-dependent. Quantitative agreement is obtained between the predictions of this band model for CdS and the edge-emission polarization experiments of D utton ( J. Phys. Chem. Solids 6 , 101 (1958)). It is shown that the spectra of edge-emission cannot be reasonably explained without the introduction of impurities or surfaces to absorb crystal momentum. In CdS, recombination from a shallow trap seems necessary to explain the large coupling to the lattice apparent in the observed emission spectrum. The edge-emission spectrum should approximate a Poisson distribution for recombination from a trap. The mean number of emitted phonons is a measure of the radius of the trapped carrier-wave function.

Allosteric interpretation of haemoglobin properties.

It is our purpose to review recent experiments on haemoglobin in order to discuss them in terms of the two state model of cooperativity. Excellent previous reviews are available of the chemistry of haemoglobin (Antonini & Brunori, 1971; Gibson, 1959 b ) which are referred to when possible. The plethora of data necessitates that a selection must be made in a review. An intentionally wide range of experiments is selected to exhibit

Relation between structure, co-operativity and spectra in a model of hemoglobin action.

Abstract A quantitative understanding of co-operativity in hemoglobin must include a description of where and how the free energy of co-operation is stored in the molecule. Experiments to date have not succeeded in associating this energy with a particular bond. One extreme possibility is that the free energy is stored as small amounts of strain energy in many bonds, so that all bonds are almost normal. A linear distributed energy model can be constructed when the strain energies of bonds are sufficiently small. This model relates quantitatively small structural changes at the heme group, upon ligation, to the free energy of cooperation, although the energy of co-operation is not chiefly stored at the heme group in this model. It is in accord with the non-co-operativity of the oxidation of hemoglobin2+ → aquo-methemoglobin3+ at pH 6, and describes the magnitude of changes observed in the Soret band when the quaternary structure is changed. It accounts for the scale of the motions in the heme region due to changes in the quaternary structure, and provides a framework for discussion of spectral and structural changes for affinity affectors which are not located near the heme group. The general model should be of use for discussing other cases of the control of local chemical properties by proteins.

An allosteric model of hemoglobin: I, kinetics

Abstract The allosteric model of Monod, Wyman & Changeux (1965 ; the Monod model) has been used to analyze the kinetics of ligand binding to mammalian hemoglobins. Combination velocity constants and dissociation velocity constants for the T and R forms were evaluated from the early measurements by Roughton & Gibson of the oxygen and carbon monoxide binding to sheep hemoglobin at pH 9.1 and more recent measurements of oxygen binding to human hemoglobin at pH 7. The difference in ligand affinities between the T and R forms, which is a factor of several hundred, is taken up almost equally by the combination and dissociation rates. The allosteric model is shown to be particularly suitable for explaining the flash photodissociation of carboxyhemoglobin. The monotonic decrease of the fraction of quickly reacting form with increasing flash intensity shown by Antonini, Chiancone & Brunori (1967) and the flow-flash experiments of Gibson & Parkhurst (1968) are both in excellent agreement with the calculations on the Monod model, with no adjustable parameters except for the fit to the equilibrium curves. In flow experiments, both the forward progess curves and the reverse dissociation curves after mixing with dithionite are very well described by the Monod model. Finally, Gibsons recent flow experiments on human hemoglobin at pH 7 have been shown to be consistent with the parameters of the Monod model which can describe the other kinetic experiments. From all this, we conclude that the kinetic experiments discussed can be fitted by the simple Monod allosteric model to well within the range of experimental errors, and that the kinetic data now available are not inconsistent with this model.