# Hartline-Ratliff model

Post-publication activity

Curator: Bruce W. Knight

The Hartline-Ratliff model represents the accurate quantitative solution of a living neural network.

## Introduction and early history

Currently (2010) the Hartline-Ratliff model, of dynamics in the thousand-faceted retina of the horseshoe crab Limulus, presents the only example of a fully characterized and solved large neural network, in an actual living species, which yields accurate and insightful prediction of that network's response to fairly arbitrary stimuli. This is achieved with a combination of standard advanced methods of dynamical system analysis.
The Hartline-Ratliff model was influential over the time-span of its publication. It gave the neuroscience community a first demonstration of a neural subsystem actually transforming input through the interaction of excitatory and inhibitory effects. It also demonstrated the potential utility of applied mathematics in the area of neurophysiology.
The early history of this study is documented in the volume edited by Ratliff (1974). (Included is Hartline's summary Nobel Lecture of 1967.) At the start of the 1930s, the compound eye of Limulus was identified by Keffer Hartline as a particularly convenient model system for the study of visual neurophysiology. The earliest single-cell recordings from visual neurons were done in Limulus and reported by Hartline and Graham (1932). In the following years a quantitative picture of Limulus vision took form. In 1949 Hartline (1949) reported the presence of a lateral inhibitory interaction between nearby Limulus retinal neurons. By 1958, quantitative measurements had advanced to the point that Hartline and Ratliff (1958) were able to present a set of equations which summarized, in the steady state, the total effect of all lateral interactions upon the firing-rate output of every given facet of the eye.

The mathematical analysis, in the steady state, of the Hartline-Ratliff model of the Limulus retina, gives clear guidance for the solution of the more challenging case of full dynamical response. We present this steady state analysis before returning to a brief discussion of that retina's physiology. If the visual neuron behind one facet (the $$m^{th}$$) fires at a rate $$e_m$$ when it alone is illuminated, then the effect of illuminating further facets is to reduce its firing rate to

$\tag{1} r_{m} =e_{m} -\sum_{n}k_{mn} r_{n}$

where the remaining $$r_n$$ are the firing rates at the other facets and the constants $$k_{mn}$$ we may reasonably call inhibitory interaction coefficients. The Equation (1) set has become known as the "Hartline-Ratliff equations".

These equations are manifestly linear, in terms of their inputs $$e_m$$ and outputs $$r_m\ ,$$ a striking feature which was experimentally confirmed in many ways prior to their publication. As a matter of mathematical convenience, an excellent approximation has proved to be the replacement of the discrete representation of (1) by a continuous one, in which a point located on the retina is indexed by a vector $$\textbf{x}$$ in two dimensions:

$\tag{2} r\left({\textbf x}\right)=e\left({\textbf x}\right)-\iint d^{2} {\textbf x}'K({\textbf x-}{\textbf x}')r\left({\textbf x}'\right).$

The dependence of the inhibitory interaction $$K$$ only upon the difference in location follows from the retina's invariant modular construction, and was experimentally verified by Barlow (1969), (see also Barlow (1967) (Thesis, 198 pages)) whose measurements also showed that it has the shape of a "volcano with a central crater, symmetric about any line through its center". A 3-dimensional perspective graph of $$-K$$ is shown in Barlow (1969) Figure 4 on page 8, or Figure (3.6) of Barlow's thesis.
The inhibitory interactive kernel $$K$$ in (2) acts upon its input $$r$$ in exactly the same way as does the point-spread function which arises in the study of image processing. This suggests that we mimic the practice in image processing, and follow an analysis in terms of the corresponding "spatial modulation transfer function"

$\tag{3} \tilde{K}({\textbf q})=\iint d^{2} {\textbf x}K\left({\textbf x}\right)\exp \left(-i{\textbf q}\cdot {\textbf x}\right).$

So if in (2) we assume the special form $\tag{4} e\left({\textbf x}\right)=e_{ {\textbf q} } \exp \left(i{\textbf q}\cdot {\textbf x}\right),$

which is a sinusoidal wave with spatial modulation frequency $$\left|{\textbf q}\right|/2\pi,$$ and if we guess that this input will lead to a proportional output $\tag{5} r\left({\textbf x}\right)=r_{ {\textbf q} } \exp (i{\textbf q}\cdot {\textbf x}),$

substitution of these two forms into (2) confirms our conjecture, as the exponential factors out and leaves $\tag{6} r_{ {\textbf q}} =e_{ {\textbf q} } -\tilde{K}\left({\textbf q}\right)r_{{\textbf q} }.$

(The simple form of this equation serves as a prototype for the more elaborate dynamical equations, such as (12), (13), and (14) below.) This we may solve for the response amplitude $\tag{7} r_{ {\textbf q}} =\frac{1}{1+\tilde{K}\left({\textbf q}\right)} e_{{\textbf q} },$

whence (multiplying both sides by the exponential) we see that $\tag{8} r\left({\textbf x}\right)=\exp \left(i{\textbf q}\cdot {\textbf x}\right)\frac{1}{1+\tilde{K}\left({\textbf q}\right)} e_{ {\textbf q} }.$

It is worth observing, at this point, that the specialization to a sinusoidal input at (4) reduced the problem to equations which, in form, no longer had an explicit spatial dependence and which were consequently tractable for solution at (6). In that equation all the dependence on spatial structure resides in a single multiplicative factor whose value depends on the wavenumber parameter. Also note that (2) expresses inhibition of response at one point in terms of feedback from responses at other points. The manifestation of feedback as a denominator term in (8) is typical for problems of this nature.
Returning to the general case, by Fourier analysis any reasonably behaved input may be expressed as a sum of spatial frequency components

$\tag{9} e\left({\textbf x}\right)=\iint d^{2} {\textbf q}\left(\exp i{\textbf q}\cdot {\textbf x}\right)e_{ {\textbf q} } ,$

and because our system is linear and hence respects superposition, from (8) we can now explicitly write down the consequent response determined by (2) as $\tag{10} r\left({\textbf x}\right)=\iint d^{2} {\textbf q}\left(\exp i{\textbf q}\cdot {\textbf x}\right)\frac{1}{1+\tilde{K}\left({\textbf q}\right)} e_{ {\textbf q} }.$

Thus the Fourier representation (9) of arbitrary input and the spatial modulation transfer function (3) enable us to solve the Hartline-Ratliff equations (2) for the steady-state output which is sculptured from arbitrary input by the inhibitory interactions.

The linear form of the Hartline-Ratliff equations (1) is definitely not a simple general consequence of the Limulus retina's underlying electrophysiology, which conforms to active-membrane nonlinear dynamical relationships as described by Hodgkin and Huxley. Rather it appears that nature has somehow overcome those nonlinearities, and has managed to craft a linear-responding device because the pressures of survival favored that outcome, presumably for reasons similar to those for the importance of linearity in the electronic reproduction of speech or music. This observation, together with the confirmed linearity of input-output in the steady state (1) suggests that we should also find linearity in the full dynamics of input-output.

## Anatomy and electrophysiology

The anatomical layout of the Limulus retina neural network has been investigated in detail. Likewise the locations, where transductions are preformed on its sensory input, have been determined by electrophysiological measurements. If, in addition, we may assume that those transductions are linear, then we have in fact enough knowledge to first devise experiments which will quantify those transductions and second to furnish a fully solved mathematical model of this retina's input-output dynamics.
Figure 1: Micrograph by William H. Miller of a cross-section through the compound eye of the horseshoe-crab Limulus, with superimposed schematic showing information flow paths and the 3 transductions: 1) light-to-voltage, 2) voltage-to-spike-rate, 3) spike-rate-to-voltage. See nearby text. Further detail in Figure 1.
Figure 1 is a micrograph, by William H. Miller, of a slice through the retina of the horseshoe crab, which has been silver-stained to reveal at the top the individual light-sensitive facets of the eye, below them some of the lateral nerve fibers which are responsible for inhibitory interactions between facets, and at the bottom optic nerve fibers on their way to the brain. Superimposed is a schematic representation of information flow. The function of the eye depends on 3 different sorts of information transducers. At the top in the box marked "1" the black object is a visual cell which contains, within an insulating membrane, biophysical machinery of molecular size which produces voltage in response to light. This signal proceeds downward within a nerve fiber (stained black in the figure), a narrow tube of electrically conducting fluid bounded by insulating membrane. It leads to the second transducer which is in the region marked "2". There the input voltage generates a train of nerve impulses which proceed (downward in the picture) along the optic nerve to the creature's brain. The rate of impulse generation at "2" is modulated by the level of voltage there.
The box marked "3" is a convergence point for lateral connections from other visual cells' impulse generators, and at "3" this merged impulse traffic modulates another (inhibitory) output voltage which likewise is fed into the impulse-train generator at "2".

## Dynamic response

It is very natural and convenient to address the topic of linear temporal system response in a way that is very similar to what we have done above for linear spatial image processing, and in the present case where both arise together, we can address them together with a single mathematical machinery. A general signal which changes both in 2-dimensional space and in time always may be expressed (following what we did above) as a weighted superposition of sinusoidal plane-waves oscillating sinusoidally in time, which individually are of the form (by generalizing (4) and (5) derivations):

$\tag{11} S({\textbf x},t)=S_{\textbf{q}, \; \omega} \cdot \left(\left( \exp i{\textbf q} \cdot \textbf{x} \right) \left( \exp i \omega t \right) \right).$

The coefficient $$S_{\textbf{q}, \; \omega}$$ is a complex number which may be expressed in terms of an amplitude and a phase. The passage of such a signal, through a linear spatiotemporal transducer whose action is homogeneous over both space and time, will produce a sinusoidal output with altered amplitude and phase, but with the same spatial wave number $$\textbf{q}$$ and temporal frequency $$\omega$$ as were input. So for such a signal, analysis of its passage through linked spatiotemporal transducers may be reduced to the algebra of complex numbers.

As suggested by Figure 1, there is a whole armory of different experimental ways from which we may deduce these complex numbers by confronting our network with input signals which are sinusoidally modulated in time or in space or both. Box "1" we may drive with modulated light, and through a microelectrode, measure the amplitude and phase of the resulting modulated voltage response. Or at box "1" we can drive current through the microelectrode to force a known modulated voltage, and measure the resulting modulated firing-rate response of box "2". Or we can alternatively measure the responding modulation in the firing rate of box "2" when we induce backward-directed modulated firing in all the other optic nerve fibers, thereby forcing the modulated rate-to-voltage transduction in box "3".
Once experiments have verified that the input-output dynamics of this retina indeed respects linear superposition, we may proceed to assemble the pieces we have collected in the laboratory, and we may construct a complete dynamical model.
As shown in Figure 1, the Limulus eye may be regarded as a collection of three such homogeneous spatial and temporal signal transducers. To enable algebraic manipulation, in Figure 2 we assign letters to both the several transduced signals and to their spatial and temporal transducers. If the stimulating input light signal $$I(\textbf{x},t)$$ is in the moving-wave form of (11), then so will be its transduced voltage signal $$v_{1}(\textbf{x},t)$$ from box "1", as will be the firing-rate induced feedback voltage signal $$v_{3}(\textbf{x},t)$$ from box "3", and likewise the output rate signal $$r(\textbf{x},t)\ .$$ Similarly to what occurred in the steady-state exercise above, with this particular choice of input all explicit dependence upon position and time is replaced by simply occurring coefficients which may, however, show complicated dependencies on the wavenumber $$\textbf{q}$$ and on the frequency $$\omega\ .$$
Figure 2: Information flow signals and the spatial and temporal transductions, given an input signal (and consequently likewise internal signals) as in (11).
In Figure 2, boxes "1" and "3" of Figure 1 are each replaced by two boxes which label the consecutive action of spatial followed by temporal transduction. Following the arrows in Figure 2, the input signal $$I$$ first encounters the Limulus eye's lens optics which somewhat blur it with a point-spread function whose corresponding spatial-modulation transfer function we call $$\tilde{P}\left({\textbf q}\right)\ .$$ The signal then passes through the light-to-voltage "generator potential" transduction which multiplies it by the laboratory-measured $$G(\omega)$$ (amplitude and phase) giving rise to the output voltage $$v_{1}\ .$$ This in turn goes to the laboratory-measured (amplitude and phase) voltage-to-rate transduction $$E(\omega)$$ (box "2") and thereafter the firing rate $$r$$ is propagated down the optic nerve. But this rate signal also is spatially distributed, through collateral branches of the optic nerve-fiber to rate-to-voltage transduction points, with different weightings at different distances, which feature was analyzed above in the discussion of the steady state. The total, convergent at one point, is given as above by multiplying with the spatial-modulation transfer-function $$-\tilde{K}(\textbf{q})\ .$$ (The negative sign here traces all way back to (1), and recognizes the important feature that the lateral interaction is inhibitory.) This signal then passes through the rate-to-voltage transduction (amplitude and phase measured in the laboratory) which here we call $$T_{L}(\omega)\ ,$$ giving rise to the box "3" output signal $$v_{3}\ .$$
Finally, the signal $$v_3$$ is fed into the voltage to spike-rate transducer (box "2" with its transduction $$E(\omega)$$) where it acts additively with the signal $$v_1\ .$$
Working back from the optic nerve in Figure 1 and Figure 2 we see that our quantitative model may be assembled from the following equations:

$\tag{12} r=E\left(\omega \right)\left(v_{1} +v_{3} \right)$

$\tag{13} v_{1} =G\left(\omega \right)\tilde{P}\left({\textbf q}\right)I$

$\tag{14} v_{3} =-T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)r.$

(The somewhat disorderly choice of symbols here has been done in an attempt to minimize notational departure from a large number of publications which were addressed, over a span of years, to different features of this program.)

Equations from (12) to (14) can be reduced to a single relationship between input $$I$$ and output $$r\ ,$$ by substituting the expressions for $$v_1$$ and $$v_3$$ into (12):

$\tag{15} r=E\left(\omega \right)\left(G\left(\omega \right)\tilde{P}\left({\textbf q}\right)I-T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)r\right).$

Solving:

$\tag{16} \left(1+E\left(\omega \right)T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)\right)r=E\left(\omega \right)G\left(\omega \right)P\left({\textbf q}\right)I$

$\tag{17} r=\frac{E\left(\omega \right)G\left(\omega \right)\tilde{P}\left({\textbf q}\right)}{1+E\left(\omega \right)T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)} I.$

Thus the spatiotemporal pattern of firing-rate $$r$$ in the optic nerve may be derived from the oscillating plane-wave input image $$I$$ by $\tag{18} r=F\left({\textbf q},\omega \right)I$

where the spatiotemporal transfer-function $$F\left({\textbf q},\omega \right)\ ,$$ a complex number, is given explicitly (from laboratory measurements) by $\tag{19} F\left({\textbf q},\omega \right)=\frac{E\left(\omega \right)G\left(\omega \right)\tilde{P}\left({\textbf q}\right)}{1+E\left(\omega \right)T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)}.$

(Again, the denominator which appears in (19) is the algebraic consequence of feedback from recurrent neuronal connections.)

We now observe (much as we did before in (9) above) that any reasonable spatiotemporal image $$I\left({\textbf x},t\right)$$ may be Fourier-analyzed in the form $\tag{20} I\left({\textbf x},t\right)=\int d\omega \iint d^{2} {\textbf q} I_{{\textbf q},\omega } \cdot \left(\left(\exp i{\textbf q}\cdot {\textbf x}\right)\left(\exp i\omega t\right)\right)$

from which, by (18) and by linear superposition, the consequent response in the optic nerve is $\tag{21} r\left({\textbf x},t\right)=\int d\omega \iint d^{2} {\textbf q}F\left({\textbf q},\omega \right)I_{{\textbf q},\omega } \cdot \left(\left(\exp i{\textbf q}\cdot {\textbf x}\right)\left(\exp i\omega t\right)\right).$

Figure 3: Response of horseshoe-crab compound eye neural network to a left-moving step in light intensity; theoretical prediction (left) and experiment (right). Middle columns: response in central region of the eye as predicted by direct use of (21). Left and right columns: response at left and right edges of illuminated area, as predicted by Wiener-Hopf extension of the theory of (21).

If, in the laboratory, we have a temporally changing pattern of light, we may computationally resolve it into a Fourier superposition as in (20), from which the firing-rate response in the optic nerve may be predicted by (21), and this may be compared to what we actually observe in the laboratory. Figure 3 (Knight (1984), after Sirovich et al. (1979)) shows the result of such a comparison. An image with a step in brightness has been moved from right to left across the eye, at 4 different velocities. Theoretical prediction, in which measured amplitude and phase responses to sinusoidal input were substituted into (19) and (21), is shown in the left set of frames and the actual outcome of response to a moving step, in the right set. For the central column of responses the calculation has been done as in (21). In the flanking columns an extended application is shown. The same laboratory-determined parameter values have been used but the theory has been somewhat enlarged (by the so-called "Wiener-Hopf technique" see (Sirovich et al. (1979)) to predict the response of a visual receptor at a boundary where the image terminates in darkness.

The dynamical Hartline-Ratliff model manifestly succeeds in these predictions. Figure 3 shows one of a very large number of successful comparisons between theory and experiment.

The middle column on either side of Figure 3 illustrates the profound transformations that the retina's activity has executed upon the image which stimulated it. That image was a step in illumination moving to the left, which resulted in a co-moving profile of neural firing. But in each case the firing profile is far different from a step, and in comparison to its most vigorous response features, the eventual steady-state elevation in firing rate, toward the right side, is barely perceptible in consequence of the near-canceling effect of lateral inhibition. Clearly the retinal circuitry is constructed to emphasize change. This is a theme now widely recognized in other nervous systems.
Early development of the dynamical model, advanced by comparison with experimental data, appears in Hartline, Ratliff, and Miller (1961), Ratliff, Hartline, and Miller (1963), Purple and Dodge (1965), Ratliff, Hartline, and Lange (1966), Lange, Hartline, and Ratliff (1966), Ratliff, Knight, Toyoda, and Hartline (1967), Ratliff, Knight, and Graham (1969), Ratliff, Knight, and Milkman (1970), Knight, Toyoda, and Dodge (1970), Shapley (1970) (Thesis, 124 pages), Shapley (1971a), Shapley (1971b), Knight (1972). A review of this early work, for non-specialists, is in Knight (1973).
The program in its mature phase, with some emphasis on spatial aspects, is presented in Brodie, Knight, and Ratliff (1978a), Brodie et al. (1978b), Sirovich, Brodie, and Knight (1979), Knight, Brodie, and Sirovich (1979). An extensive discussion is in Brodie's thesis (1979. 328 pages). (It was Dr. Brodie who reasonably suggested that the overall dynamical model be called the Hartline-Ratliff Model.) A concise retrospective review of the model is given in Knight (1984, section 2: Background, 8 pages).
One well might ask whether any broadly applicable lessons may be learned from this study. The Hartline-Ratliff dynamical model has been notably successful in the quantitative prediction of the neural responses which it was constructed to address. Also the theoretical structure of the model is quite different from that of models which have arisen since to account for the responses of other neural networks.
Clearly the model's creation depended on the choice of a particularly simple system which could be characterized in all necessary detail and which also conveniently permitted extensive manipulation in the laboratory. As shown in Figure 1, the Limulus eye is constructed from only one type of impulse-generating neuron; the transductions "1" and "2" are within a single cell and the post-synaptic transduction "3" is between cells of a single dynamical design. The individual components are relatively quite large (the facets in Figure 1 measure 2/10 of a millimeter) which allows convenient micromanipulation under a microscope which has a large depth-of-field. The preparation is very robust; when removed from the creature it continues to function well for many hours, and easily tolerates the necessary quite elaborate microsurgical procedures.
All the above were necessary for the experiments which first led to (1). The great simplicity of that linear relationship, in spite of what known electrophysiology would lead one to anticipate, suggested that linearity was enforced by survival need, and might also surface in the detailed dynamics as well.
The theoretical structure of the Hartline-Ratliff dynamical model, which is expressed in terms of linear system analysis, is a direct consequence of experimentally confirming that linearity. Experimental quantification of the model also depended on that linearity, which allowed measurements made with stimuli scaled for experimental convenience, to determine responses at other stimulus scales without the need for further experiment.
The Hartline-Ratliff model is unusual in that plausible electrophysiology plays no quantitative role in its construction. The remarkable appearance of linearity enables us to jump from input to output without having to address the intermediating mechanism. Nonetheless, the nature of the electrophysiology which achieves this is of great potential interest and hopefully might be revealed in the future.
Neural networks under study in other real organisms tend to involve interactions among several inequivalent neuron types. That precludes the trial of a simple linearity conjecture as in (1). But it is not hard to propose conjectural multi-neuron-type relations, similar to (1) in the sense that they might enhance a neural network's abilities both to perform and to evolve. So far, the Hartline-Ratliff model is the only instance where such conjecture has been pursued, and the model has been confirmed and has yielded rewards. Since then laboratory technique has come a long way, and a quest for somewhat similar deep structure in another system again might prove quite rewarding.
Here the urge is irresistible to mention that, since the inception of the Hartline-Ratliff model, the observation has arisen that plausible neuron electrophysiology includes a subset of designs for which response linearity appears naturally as an emergent property when an assembly of such neurons perform together as a population. See Knight (2008), particularly appendix 2.

## Acknowledgments

The author would like to extend thanks for much help with the manuscript to Carol Feltes, Mary Margaret Hickey, Jeanine McSweeney, and Ellen Paley.

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