# Hartline-Ratliff model

Bruce W. Knight (2011), Scholarpedia, 6(1):2121. | doi:10.4249/scholarpedia.2121 | revision #121375 [link to/cite this article] |

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

## Contents |

## Introduction and early history

## The steady state

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

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".

*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:

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

*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.

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

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 \[\tag{8} r\left({\textbf x}\right)=\exp \left(i{\textbf q}\cdot {\textbf x}\right)\frac{1}{1+\tilde{K}\left({\textbf q}\right)} e_{ {\textbf q} }. \]

*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.

## Anatomy and electrophysiology

## Dynamic response

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

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.

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

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

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

(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.)

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

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 \[\tag{21} r\left({\textbf x},t\right)=\int d\omega \iint d^{2} {\textbf q}F\left({\textbf q},\omega \right)I_{{\textbf q},\omega } \cdot \left(\left(\exp i{\textbf q}\cdot {\textbf x}\right)\left(\exp i\omega t\right)\right). \]

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.

## Comments

*Hartline-Ratliff Model*.) A concise retrospective review of the model is given in Knight (1984, section 2: Background, 8 pages).

## Acknowledgments

## Bibliography

- Barlow, R.B. Jr. (1967) Inhibitory fields in the Limulus lateral eye. Thesis, The Rockefeller University, New York. (online)
- Barlow, R.B. Jr. (1969) Inhibitory field in the Limulus eye. J. Gen. Physiol. 54, 383-396. (online)
- Brodie, S.E., Knight, B.W., and Ratliff, F. (1978) The Response of the Limulus Retina to moving stimuli: A prediction by Fourier Synthesis. J. Gen. Physiol. 72, 129-166. (online)
- Brodie, S.E., Knight, B.W., and Ratliff, F. (1978) The spatiotemporal transfer function of the Limulus lateral eye. J. Gen. Physiol. 72, 167-202. (online)
- Brodie, S.E. (1979) Analysis and synthesis of the dynamic response of retinal neurons. Thesis, The Rockefeller University, New York. (online)
- Hartline, H.K. and Graham, C.H. (1932) Nerve impulses from single receptors in the eye. Journal of Cellular and Comparative Physiology, 1(2), 277-295. (online)
- Hartline, H.K. (1949) Inhibition of activity of visual receptors by illuminating nearby retinal areas in the Limulus eye. Federation Proceedings, 8(1), 69-69. (online)
- Hartline, H.K., Ratliff, F., Miller, W.H. (1961) Inhibitory interaction in the retina and its significance in vision. In Nervous Inhibition, E. Florey, ed., 241-284, Pergamon Press, London. (online)
- Hartline, H.K., Ratliff, F. (1958) Spatial summation of inhibitory influences in the eye of Limulus, and the mutual interaction of receptor units. J. Gen. Physiol. 41, 1049-1066. (online)
- Hartline, H.K. (1967) Visual receptors and retinal interaction. Les Prix Nobel en 1967. Nobel Foundation 1968, 1969, 242-259. (online)
- Knight, B.W., Toyoda, J., Dodge, F.A. Jr. (1970) A quantitative description of the dynamics of excitation and inhibition in the eye of Limulus. J. Gen. Physiol. 56, 421-437 (online)
- Knight, B.W. (1972) Dynamics of encoding in a population of neurons. J. Gen. Physiol. 59, 734-766. (online)
- Knight, B.W., Jr. (1973) The Horseshoe Crab Eye: A little nervous system that is solvable. In: Lectures on Mathematics in the Life Sciences, Some Mathematical Questions in Biology IV. J.B. Cowen (Ed.), Am. Math. Soc. 5, 113-144. (online)
- Knight, B.W., Jr., Brodie, S.E. and Sirovich, L. (1979) Treatment of nerve impulse data for comparison with theory, PNAS (USA) 76, 6026-6029. (online)
- Knight, B.W., Jr. (1984) How Hamiltonian dynamical theory in the complex domain yields asymptotic solutions to the non-Hermitian integral equations of visual nerve-networks. In Mathematical Physics VII. Eds: Brittin, W.E.; Gustafson, K.; Wyss, W. p. 431-453. (online)
- Knight, B.W. (2008) Some hidden physiology in naturalistic spike rasters. The faithful copy neuron. Brain Connectivity Workshop, Sydney. (online)
- Lange, D., Hartline, H.K., Ratliff, F. (1966) The dynamics of lateral inhibition in the compound eye of Limulus. II In The Functional Organization of the Compound Eye, 425-449 Pergamon Press, London. (online)
- Purple, L. and Dodge, F.A. (1965) Interaction of excitation and inhibition in the eccentric cell in the eye of Limulus. Cold Spring Harbor Symposia on Quantitative Biology 30, 529-537. (online)
- Ratliff, F. and Hartline, H.K. (1959) The responses of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic. Journal of General Physiology, 42, 1241-1255. (online)
- Ratliff, F., Hartline, H.K., and Miller, W.H. (1963) Spatial and temporal aspects of retinal inhibitory interaction. Journal of the Optical Society of America, 53, 110-120. (online)
- Ratliff, F., Hartline, H.K., Lange, D. (1966) The dynamics of lateral inhibition in the compound eye of Limulus. I. In The Functional Organization of the Compound Eye, 399-424 Pergamon Press, London. (online)
- Ratliff, F., Knight, B.W., Toyoda, J., Hartline, H.K. (1967) Enhancement of flicker by lateral inhibition. Science. 158, 392-393. (online)
- Ratliff, F., Knight, B.W., and Graham, N. (1969) On Tuning and Amplification by Lateral Inhibition. PNAS (USA) 62, 733-740. (online)
- Ratliff, F., Knight, B.W., Milkman, N. (1970) Superposition of excitatory and inhibitory influences in the retina of Limulus: effect of delayed inhibition. PNAS, 67 1558-1564. (online)
- Ratliff, F. (1974) Studies on excitation and inhibition in the retina: A collection of papers from the Laboratories of H. Keffer Hartline. NY: Rockefeller University Press. (online)
- Shapley, R. (1970) Variability in the firing of nerve impulses in eccentric cells of the Limulus eye. Thesis. The Rockefeller University, New York. (online)
- Shapley, R. (1971) Fluctuations of the impulse rate in Limulus eccentric cells. J. Gen. Physiol. 57, 539-556. (online)
- Shapley, R. (1971) Effects of lateral inhibition on fluctuations of the impulse rate. J. Gen. Physiol. 57, 557-575. (online)
- Sirovich, L., Brodie, S.E. and Knight, B. W., Jr. (1979) Effect of boundaries on the response of a neural network. Biophys. J. 28, 423-446. (online)

## Further reading

- Knight, B.W., Jr. (1973) The Horseshoe Crab Eye: A little nervous system that is solvable. In: Lectures on Mathematics in the Life Sciences, Some Mathematical Questions in Biology IV. J.B. Cowen (Ed.), Am. Math. Soc. 5, 113-144. (online)
- Knight, B.W., Jr. (1984) How Hamiltonian dynamical theory in the complex domain yields asymptotic solutions to the non-Hermitian integral equations of visual nerve-networks. In Mathematical Physics VII. Eds: Brittin, W.E.; Gustafson, K.; Wyss, W. p. 431-453. (online)
- Knight, B.W. (2008) Some hidden physiology in naturalistic spike rasters. The faithful copy neuron. Brain Connectivity Workshop, Sydney. (online)

## External links

- Bruce Knight's page at the Rockefeller University
- Laboratory of Biophysics at the Rockefeller University