doi: 10.1371/journal.pone.0077395.
eCollection 2013.
Optimal design for hetero-associative memory: hippocampal CA1 phase response curve and spike-timing-dependent plasticity
Affiliations
- PMID: 24204822
- PMCID: PMC3812027
- DOI: 10.1371/journal.pone.0077395
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Optimal design for hetero-associative memory: hippocampal CA1 phase response curve and spike-timing-dependent plasticity
PLoS One.
.
Abstract
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.
Conflict of interest statement
Figures
We derive pairs of PRCs and STDP window functions optimally recalling normal, reversed, and spread-out memory spike patterns.
(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 in vitro
, . 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. Left: Symmetric plasticity rule . Right: Asymmetric plasticity rule .
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 in vitro
, . 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. Left: Symmetric plasticity rule . Right: Asymmetric plasticity rule .
We use a typical STDP window function (left panel of Fig. 2C
[16]) and the PRC (cell 
1 in Fig. 2B
[18]) 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).
1 in Fig. 2B
[18]) 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).
(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 Fig. 2B. 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.
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 Fig. 2B. 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.
We computed the Fourier series of symmetric and asymmetric STDP window functions in Fig. 2C and compared the first two frequency components of the STDP window functions in Fig. 2C with those in Figs. 4A′–D′. (A) Symmetric and asymmetric STDP window functions composed of only the fundamental and second frequency components of the ones in Fig. 2C. Left: Symmetric plasticity rule . Right: Asymmetric plasticity rule . (B) Rates of fundamental and second frequency components of STDP window functions in Fig. 5A and the purposely designed ones in Figs. 4A′–D′. We compared the amplitudes between the two Fourier coefficients of each STDP window function, i.e., 
and 
. Symmetric: left panel of Fig. 5A
. Asymmetric: right panel of Fig. 5A
.
and . Symmetric: left panel of Fig. 5A
. Asymmetric: right panel of Fig. 5A
.
The synaptic weight 
was determined using the STDP window function (cell 
5 in Fig. 4A′) 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 Fig. 2B) was verified by using numerical simulations (
, 
, 
). (A) Normal spike pattern retrieval (
). (B) Reversed pattern retrieval (
). (C) Doubly spread-out pattern retrieval (
). Left column: Time evolution of the amplitude of the overlap between the 
-th frequency component of 
and the 
-th frequency component of 
, 
. Center column: An example of the memory output pattern as originally stored, 
. Right column: The retrieval output pattern 
at equilibrium (corresponding to 
in left column).
was determined using the STDP window function (cell 5 in Fig. 4A′) 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 Fig. 2B) was verified by using numerical simulations (, , ). (A) Normal spike pattern retrieval (). (B) Reversed pattern retrieval (). (C) Doubly spread-out pattern retrieval (). Left column: Time evolution of the amplitude of the overlap between the -th frequency component of and the -th frequency component of , . Center column: An example of the memory output pattern as originally stored, . Right column: The retrieval output pattern at equilibrium (corresponding to in left column).Similar articles
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Grants and funding
This work was partially supported by a Grant-in-Aid for Japan Society for the Promotion of Science Fellows [No. 12J09230 (RM)] and a Grant-in-Aid for Scientific Research (C) [No. 23500375 (TA)] from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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