Optimal Design for Hetero-Associative Memory: Hippocampal CA1 Phase Response Curve and Spike-Timing-Dependent Plasticity

<div><p>Recently reported experimental findings suggest that the hippocampal CA1 network stores spatio-temporal spike patterns and retrieves temporally reversed and spread-out patterns. In this paper, we explore the idea that the properties of the neural interactions and the synaptic plasticity rule in the CA1 network enable it to function as a hetero-associative memory recalling such reversed and spread-out spike patterns. In line with Lengyel’s speculation (Lengyel et al., 2005), we firstly derive optimally designed spike-timing-dependent plasticity (STDP) rules that are matched to neural interactions formalized in terms of phase response curves (PRCs) for performing the hetero-associative memory function. By maximizing object functions formulated in terms of mutual information for evaluating memory retrieval performance, we search for STDP window functions that are optimal for retrieval of normal and doubly spread-out patterns under the constraint that the PRCs are those of CA1 pyramidal neurons. The system, which can retrieve normal and doubly spread-out patterns, can also retrieve reversed patterns with the same quality. Finally, we demonstrate that purposely designed STDP window functions qualitatively conform to typical ones found in CA1 pyramidal neurons.</p></div

Comparison of purposely designed STDP window functions (Figs. 4A′–D′) and those reported for the hippocampal CA1 region.

<p>We computed the Fourier series of symmetric and asymmetric STDP window functions in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2C</a> and compared the first two frequency components of the STDP window functions in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2C</a> with those in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g004" target="_blank">Figs. 4A′–D′</a>. (A) Symmetric and asymmetric STDP window functions composed of only the fundamental and second frequency components of the ones in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2C</a>. <i>Left</i>: Symmetric plasticity rule <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Wittenberg1" target="_blank">[16]</a>. <i>Right</i>: Asymmetric plasticity rule <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Bi1" target="_blank">[17]</a>. (B) Rates of fundamental and second frequency components of STDP window functions in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g005" target="_blank">Fig. 5A</a> and the purposely designed ones in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g004" target="_blank">Figs. 4A′–D′</a>. We compared the amplitudes between the two Fourier coefficients of each STDP window function, i.e., and . Symmetric: <i>left</i> panel of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g005" target="_blank">Fig. 5A </a><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Wittenberg1" target="_blank">[16]</a>. Asymmetric: <i>right</i> panel of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g005" target="_blank">Fig. 5A </a><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Bi1" target="_blank">[17]</a>.</p

Structure of hetero-associative memory model.

<p>(A) Schematic diagram of a feedforward network with neural oscillators. Presynaptic neurons numbered are characterized by their initial phases, , representing their individual spiking timings. The angle of the radius line in the circle represents the initial phase. Postsynaptic neurons numbered are characterized by their initial phases, . The pre- and postsynaptic neurons are fully connected by synaptic connections. (B) Phase response curves (PRCs) of hippocampal CA1 pyramidal neurons recorded <i>in vitro </i><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Ota1" target="_blank">[18]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Ota2" target="_blank">[19]</a>. The abscissa represents the phase of a perturbation arrival, and the ordinate represents the phase shift of the postsynaptic spike in response to the perturbation current. (C) Typical STDP window functions observed in hippocampal CA1 pyramidal neurons. In the storage process, synaptic weights are determined in accordance with an STDP learning rule depending on the phase difference between the pre- and postsynaptic spikes. <i>Left</i>: Symmetric plasticity rule <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Wittenberg1" target="_blank">[16]</a>. <i>Right</i>: Asymmetric plasticity rule <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Bi1" target="_blank">[17]</a>.</p

Outline of hetero-associative memory functions we studied.

<p>Phase patterns of presynaptic neurons are associated with those of postsynaptic neurons in the hetero-associative memory. In the storage process, pairs of pre- and postsynaptic phase patterns, and (), are embedded by modifying synaptic weights in accordance with an STDP learning rule. In the retrieval process, when presented with a phase pattern of presynaptic neurons which resembles the <i>μ</i>-th memory key pattern that is temporally reversed and/or stretched to times its original timescale, (), the postsynaptic neurons recall a phase pattern which resembles the associated memory output pattern that is temporally reversed and/or stretched to times its original timescale, .</p

Examples of STDP window functions optimally matched to PRCs of five hippocampal CA1 pyramidal neurons shown in Fig. 2B.

<p>(A–D) By maximizing the objective function defined in Eq. (25), we searched for STDP window functions that are optimal for retrieving normal patterns. (A′–D′) By maximizing the objective function defined in Eq. (24), we searched for ones that are optimal for both retrieving normal and doubly spread-out patterns. In all cases, , . We obtained connected sets of optimal STDP window functions, as described in the main article. Each of the four panels in the upper and lower rows plots examples of optimal STDP window functions with different phases. The numbers assigned to each line correspond to the cell indexes in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2B</a>. All sets of optimal STDP window functions except for cell #1 have the same form. (A, A′) STDP window functions when , which corresponds to the symmetric STDP rule. (B, B′) STDP window functions when (B) and (B′), which correspond to the asymmetric STDP rule. (C, C′) STDP window functions when , which corresponds to the inverted symmetric STDP rule. (D, D′) STDP window functions when (D) and (D′), which correspond to the inverted asymmetric STDP rule.</p

Performance of hetero-associative memory model with typical parameters.

<p>We use a typical STDP window function (<i>left</i> panel of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2C </a><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Wittenberg1" target="_blank">[16]</a>) and the PRC (cell 1 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2B </a><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone.0077395-Ota1" target="_blank">[18]</a>) measured from hippocampal CA1 pyramidal neurons. In this simulation, . Given a retrieval key pattern similar to , is to be retrieved (i.e., normal spike pattern retrieval). (A) Amplitudes of the overlaps ( denotes the wavenumber) at equilibrium as a function of the noise intensity when . As defined in Eq. (26), is the overlap between the first memory output pattern and the retrieval output pattern in the -th frequency component: . represents the characteristic function of the postsynaptic phase distribution at each wavenumber . Solid curves are theoretical results obtained from Eq. (27); The plotted points are from numerical simulations using LPE (13). (B) An example of the PDF (19) and a histogram of phase difference obtained by numerically solving the LPE (13) at equilibrium. and . (C) Amplitudes of the overlaps () as a function of the concentration parameter . As defined in Eq. (12), is the overlap between the first memory key pattern and the retrieval key pattern in the -th frequency component: . represents the characteristic function of the presynaptic phase distribution at each wavenumber . Solid curves are theoretical results obtained from Eq. (15); Plots are obtained from a retrieval key pattern randomly generated with the von Mises PDF (14). (D) Amplitudes of the overlaps () at equilibrium as a function of . . Solid curves are theoretical results obtained from Eq. (27); The plots are from numerical simulations using LPE (13).</p

Confirmation that the system with the STDP window functions in Figs. 4(A′–D′) can function as intended.

<p>The synaptic weight was determined using the STDP window function (cell 5 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g004" target="_blank">Fig. 4A′</a>) to store three pairs of random phase patterns, and (), and when presented with the retrieval key pattern generated with the conditional PDF (Eq. (14)) given , the retrieval performance of the system with the determined synaptic weight and the measured PRC (cell #5 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077395#pone-0077395-g002" target="_blank">Fig. 2B</a>) was verified by using numerical simulations (, , ). (A) Normal spike pattern retrieval (). (B) Reversed pattern retrieval (). (C) Doubly spread-out pattern retrieval (). <i>Left column</i>: Time evolution of the amplitude of the overlap between the -th frequency component of and the -th frequency component of , . <i>Center column</i>: An example of the memory output pattern as originally stored, . <i>Right column</i>: The retrieval output pattern at equilibrium (corresponding to in <i>left column</i>).</p

Outline of our approach.

<p>We derive pairs of PRCs and STDP window functions optimally recalling normal, reversed, and spread-out memory spike patterns.</p

Change in the theta frequency before the first lever press and after the last lever release.

<p>(A) Increase in the dominant frequency of theta oscillation before the first lever press in the correct and incorrect trials. The filled and open circles indicate the data averaged over the correct and incorrect trials in each rat, respectively, and then over all the rats. (B) Increase in the dominant frequency of theta oscillation before the first lever press (a) and decrease after the final lever off (b) in the right and left forelimb trials. The filled and open squares indicate the data averaged over the right and left trials in each rat, respectively, and then over all the rats. The error-bars indicate the standard error of the mean.</p

Change in hippocampal theta oscillation associated with multiple lever presses in a bimanual two-lever choice task for robot control in rats

<div><p>Hippocampal theta oscillations have been implicated in working memory and attentional process, which might be useful for the brain-machine interface (BMI). To further elucidate the properties of the hippocampal theta oscillations that can be used in BMI, we investigated hippocampal theta oscillations during a two-lever choice task. During the task body-restrained rats were trained with a food reward to move an e-puck robot towards them by pressing the correct lever, ipsilateral to the robot several times, using the ipsilateral forelimb. The robot carried food and moved along a semicircle track set in front of the rat. We demonstrated that the power of hippocampal theta oscillations gradually increased during a 6-s preparatory period before the start of multiple lever pressing, irrespective of whether the correct lever choice or forelimb side were used. In addition, there was a significant difference in the theta power after the first choice, between correct and incorrect trials. During the correct trials the theta power was highest during the first lever-releasing period, whereas in the incorrect trials it occurred during the second correct lever-pressing period. We also analyzed the hippocampal theta oscillations at the termination of multiple lever pressing during the correct trials. Irrespective of whether the correct forelimb side was used, the power of hippocampal theta oscillations gradually decreased with the termination of multiple lever pressing. The frequency of theta oscillation also demonstrated an increase and decrease, before and after multiple lever pressing, respectively. There was a transient increase in frequency after the first lever press during the incorrect trials, while no such increase was observed during the correct trials. These results suggested that hippocampal theta oscillations reflect some aspects of preparatory and cognitive neural activities during the robot controlling task, which could be used for BMI.</p></div