Hartline Ratliff model - davidar/scholarpedia GitHub Wiki

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

Table of Contents Introduction and early history The steady state Anatomy and electrophysiology Dynamic response Comments Acknowledgments Bibliography Further reading External links See also

<math>\label{eq1}

r_{m} =e_{m} -\sum_{n}k_{mn} r_{n} </math>

where the remaining <math>r_n</math> are the firing rates at the other facets and the constants <math>k_{mn}</math> we may reasonably call inhibitory interaction coefficients. The Equation \eqref{eq1} set has become known as the "Hartline-Ratliff equations".

<math>\label{eq2}

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). </math>

<math>\label{eq3}

\tilde{K}({\textbf q})=\iint d^{2} {\textbf x}K\left({\textbf x}\right)\exp \left(-i{\textbf q}\cdot {\textbf x}\right). </math>

So if in \eqref{eq2} we assume the special form

<math>\label{eq4}

e\left({\textbf x}\right)=e_{ {\textbf q} } \exp \left(i{\textbf q}\cdot {\textbf x}\right), </math>

which is a sinusoidal wave with spatial modulation frequency <math>\left|{\textbf q}\right|/2\pi,</math> and if we guess that this input will lead to a proportional output

<math>\label{eq5}

r\left({\textbf x}\right)=r_{ {\textbf q} } \exp (i{\textbf q}\cdot {\textbf x}), </math>

substitution of these two forms into \eqref{eq2} confirms our conjecture, as the exponential factors out and leaves

<math></math>\label{eq6}

r_{ {\textbf q}} =e_{ {\textbf q} } -\tilde{K}\left({\textbf q}\right)r_ =\frac{1}{1+\tilde{K}\left({\textbf q}\right)} e_<math>\label{eq8} 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} }. </math>

<math>\label{eq9}

e\left({\textbf x}\right)=\iint d^{2} {\textbf q}\left(\exp i{\textbf q}\cdot {\textbf x}\right)e_{ {\textbf q} } , </math>

and because our system is linear and hence respects superposition, from \eqref{eq8} we can now explicitly write down the consequent response determined by \eqref{eq2} as

<math>\label{eq10}

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} }. </math>

Thus the Fourier representation \eqref{eq9} of arbitrary input and the spatial modulation transfer function \eqref{eq3} enable us to solve the Hartline-Ratliff equations \eqref{eq2} for the steady-state output which is sculptured from arbitrary input by the inhibitory interactions.

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 Horseshoe-crab-limulus-micrograph_v2.png.

<math>\label{eq11}

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). </math>

The coefficient <math>S_{\textbf{q}, \; \omega}</math> 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 <math>\textbf{q}</math> and temporal frequency <math>\omega</math> 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.

Information flow signals and the spatial and temporal transductions, given an input signal (and consequently likewise internal signals) as in \eqref{eq11}.

<math>\label{eq12}

r=E\left(\omega \right)\left(v_{1} +v_{3} \right) </math>

<math>\label{eq13}

v_{1} =G\left(\omega \right)\tilde{P}\left({\textbf q}\right)I </math>

<math>\label{eq14}

v_{3} =-T_{L} \left(\omega \right)\tilde{K}\left({\textbf q}\right)r. </math>

(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.)

<math>\label{eq15}

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). </math>

Solving:

<math>\label{eq16}

\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 </math>

<math>\label{eq17}

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. </math>

Thus the spatiotemporal pattern of firing-rate <math>r</math> in the optic nerve may be derived from the oscillating plane-wave input image <math>I</math> by

<math>\label{eq18}

r=F\left({\textbf q},\omega \right)I </math>

where the spatiotemporal transfer-function <math>F\left({\textbf q},\omega \right)\ ,</math> a complex number, is given explicitly (from laboratory measurements) by

<math>\label{eq19}

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)}. </math>

(Again, the denominator which appears in \eqref{eq19} is the algebraic consequence of feedback from recurrent neuronal connections.)

We now observe (much as we did before in \eqref{eq9} above) that any reasonable spatiotemporal image <math>I\left({\textbf x},t\right)</math> may be Fourier-analyzed in the form

<math></math>\label{eq20}

I\left({\textbf x},t\right)=\int d\omega \iint d^{2} {\textbf q} I_<math></math>\label{eq21} r\left({\textbf x},t\right)=\int d\omega \iint d^{2} {\textbf q}F\left({\textbf q},\omega \right)I_ 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)

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

• Neural network, Electrophysiology

Category:Vision Category:Computational neuroscience