in the present study , a simplified mathematical model of the dynamic activation of pka and erk was developed ( fig .
inducer. because activation of erk and pka are both required for ltf , inducer was proportional to the product of pka and erk activities .
the peak levels of inducer , which corresponded to the peak synergistic interaction between pka and erk , were hypothesized to predict the efficacy of stimulus protocols . as a point of reference , the simulated peak level of inducer produced by five , 5-min pulses of 5-ht with uniform 20-min interstimulus intervals ( isis ) was selected ( fig .
this ' standard ' protocol has been widely used in experimental studies since its introduction in 1986 . to identify a protocol that produced the highest peak level of inducer , 9,999 alternative protocols were simulated .
each protocol included five , 5-min 5-ht stimuli , but the isis were chosen as multiples of 5 min , in the range from 550 min ( supplementary movie 1 ) .
thus , each of the four isis had 10 possible values ( i.e. , 10 total possible permutations , one of which represented the standard protocol ) .
simulation of these protocols revealed considerable variability in the peak level of inducer ( supplementary fig .
the protocol that maximized the peak level of inducer ( the ' enhanced ' protocol ) consisted of non - uniformly spaced 5-ht applications with isis of 10 , 10 , 5 , and 30 min .
1c ) . to the extent that the model captured the salient features of the dynamics of pka and erk and the extent to which the synergism between pka and erk activity is essential for ltf , the enhanced protocol was predicted to induce more ltf than the standard protocol .
three groups of preparations were examined , the standard and enhanced protocols ( see fig .
1b , c ) and a vehicle ( control ) group , which received five 5-min applications of solution without 5-ht ( 20-min isis ) . in each group , excitatory postsynaptic potentials ( epsps ) were measured prior to 5-ht treatment ( pretest ) and 1 , 2 , and 5 d after 5-ht treatment ( post - tests ) .
long - term facilitation was defined as a significant increase in the epsp amplitude in either the standard or enhanced groups as compared to epsps measured in the control group .
sample recordings are illustrated in figure 2a and summary data are illustrated in figure 2b .
statistical analyses ( two - way anova ) indicated significant overall differences in the amplitude of epsps among the three treatment groups ( f2,73 = 22.06 , p < 0.001 ) and among the three time points ( f2,73 = 7.28 , p < 0.002 ) .
post hoc , pair - wise comparisons ( student - newman - keuls tests , snk ) indicated that 1 day after 5-ht treatment both the standard protocol ( q2 = 4.49 , n = 10 , p < 0.003 ) and enhanced protocol ( q3 = 7.57 , n = 11 , p < 0.001 ) induced ltf as compared to vehicle controls ( n = 8) .
the ltf that was induced by the standard protocol , however , did not persist beyond the first day ( day 2 : q2 = 2.54 , n = 10 ; day 5 : q2 = 0.21 , n = 9 ) as compared to controls on day 2 ( n = 8) and on day 5 ( n = 8) .
in contrast , the ltf that was induced by the enhanced protocol persisted for up to five days ( day 2 : q3 = 5.21 , n = 9 , p < 0.002 ; day 5 : q2 = 3.41 , n = 9 , p < 0.02 ) as compared to vehicle controls .
in addition , post hoc analyses indicated that the enhanced protocol induced significantly greater ltf at all time points as compared to the standard protocol ( day 1 : q2 = 3.18 , p < 0.03 ; day 2 : q2 = 2.89 , p < 0.05 ; day 5 : q3 = 3.73 , p < 0.03 ) .
the induction and consolidation of ltf in vitro depends on the activity of several transcription factors .
for example , increased levels of creb1 phosphorylation are associated with the induction and consolidation of ltf at sensorimotor synapses . because the enhanced protocol induced greater and longer - lasting ltf , the enhanced protocol was hypothesized to be associated with a greater level of creb1 phosphorylation .
three groups of sensory neurons cultures were examined , one group was treated with the standard protocol , the second group was treated with the enhanced protocol identical to that used in the electrophysiological experiments of figure 2 , and the third group was a vehicle control . in each group , the level of creb1 phosphorylation was measured either immediately ( 0 h ) or 18 h after treatment , and the values in the standard and enhanced groups were normalized to vehicle controls .
statistical analyses ( two - way anova ) indicated overall significant differences between the two 5-ht protocols ( f1,18 = 30.12 , p < 0.001 ) .
pair - wise comparisons ( snk tests ) indicated that compared to the standard protocol , the enhanced protocol induced significantly higher levels of creb1 phosphorylation immediately ( q2 = 6.94 , n = 4 , p < 0.001 ) , and 18 h after treatment ( q2 = 3.69 , n = 7 , p < 0.02 ) . both long - term synaptic plasticity and modifications in creb1 are thought to be key processes underlying long - term memory .
having demonstrated that the enhanced protocol improved ltf and phosphorylation of creb1 , the enhanced protocol was hypothesized to produce stronger learning than the standard protocol in behaving animals .
4 ) . statistical analyses ( two - way anova ) indicated significant differences among the three training groups ( f2,82 = 11.77 , p < 0.001 ) .
pair - wise comparisons ( snk tests ) indicated that both the standard protocol ( q2 = 3.12 , n = 12 , p < 0.04 ) and enhanced protocol ( q3 = 4.13 , n = 14 , p < 0.02 ) induced lts that lasted 1 d. the lts that was induced by the standard protocol , however , did not persist for five days ( q2 = 0.38 , n = 14 ) .
in contrast , the lts that was induced by the enhanced protocol persisted for at least five days ( q3 = 5.48 , n = 9 , p < 0.001 ) .
in the present study , a simplified mathematical model of the dynamic activation of pka and erk was developed ( fig .
inducer. because activation of erk and pka are both required for ltf , inducer was proportional to the product of pka and erk activities .
the peak levels of inducer , which corresponded to the peak synergistic interaction between pka and erk , were hypothesized to predict the efficacy of stimulus protocols . as a point of reference , the simulated peak level of inducer produced by five , 5-min pulses of 5-ht with uniform 20-min interstimulus intervals ( isis ) was selected ( fig .
this ' standard ' protocol has been widely used in experimental studies since its introduction in 1986 . to identify a protocol that produced the highest peak level of inducer , 9,999 alternative protocols were simulated .
each protocol included five , 5-min 5-ht stimuli , but the isis were chosen as multiples of 5 min , in the range from 550 min ( supplementary movie 1 ) .
thus , each of the four isis had 10 possible values ( i.e. , 10 total possible permutations , one of which represented the standard protocol ) .
simulation of these protocols revealed considerable variability in the peak level of inducer ( supplementary fig .
the protocol that maximized the peak level of inducer ( the ' enhanced ' protocol ) consisted of non - uniformly spaced 5-ht applications with isis of 10 , 10 , 5 , and 30 min .
1c ) . to the extent that the model captured the salient features of the dynamics of pka and erk and the extent to which the synergism between pka and erk activity is essential for ltf , the enhanced protocol was predicted to induce more ltf than the standard protocol .
three groups of preparations were examined , the standard and enhanced protocols ( see fig .
1b , c ) and a vehicle ( control ) group , which received five 5-min applications of solution without 5-ht ( 20-min isis ) . in each group ,
excitatory postsynaptic potentials ( epsps ) were measured prior to 5-ht treatment ( pretest ) and 1 , 2 , and 5 d after 5-ht treatment ( post - tests ) .
long - term facilitation was defined as a significant increase in the epsp amplitude in either the standard or enhanced groups as compared to epsps measured in the control group .
sample recordings are illustrated in figure 2a and summary data are illustrated in figure 2b .
statistical analyses ( two - way anova ) indicated significant overall differences in the amplitude of epsps among the three treatment groups ( f2,73 = 22.06 , p < 0.001 ) and among the three time points ( f2,73 = 7.28 , p < 0.002 ) .
post hoc , pair - wise comparisons ( student - newman - keuls tests , snk ) indicated that 1 day after 5-ht treatment both the standard protocol ( q2 = 4.49 , n = 10 , p < 0.003 ) and enhanced protocol ( q3 = 7.57 , n = 11 , p < 0.001 ) induced ltf as compared to vehicle controls ( n = 8) .
the ltf that was induced by the standard protocol , however , did not persist beyond the first day ( day 2 : q2 = 2.54 , n = 10 ; day 5 : q2 = 0.21 , n = 9 ) as compared to controls on day 2 ( n = 8) and on day 5 ( n = 8) .
in contrast , the ltf that was induced by the enhanced protocol persisted for up to five days ( day 2 : q3 = 5.21 , n = 9 , p < 0.002 ; day 5 : q2 = 3.41 , n = 9 , p < 0.02 ) as compared to vehicle controls .
in addition , post hoc analyses indicated that the enhanced protocol induced significantly greater ltf at all time points as compared to the standard protocol ( day 1 : q2 = 3.18 , p < 0.03 ; day 2 : q2 = 2.89 , p < 0.05 ; day 5 : q3 = 3.73 , p < 0.03 ) .
the induction and consolidation of ltf in vitro depends on the activity of several transcription factors .
for example , increased levels of creb1 phosphorylation are associated with the induction and consolidation of ltf at sensorimotor synapses . because the enhanced protocol induced greater and longer - lasting ltf , the enhanced protocol was hypothesized to be associated with a greater level of creb1 phosphorylation .
three groups of sensory neurons cultures were examined , one group was treated with the standard protocol , the second group was treated with the enhanced protocol identical to that used in the electrophysiological experiments of figure 2 , and the third group was a vehicle control . in each group , the level of creb1 phosphorylation was measured either immediately ( 0 h ) or 18 h after treatment , and the values in the standard and enhanced groups were normalized to vehicle controls .
statistical analyses ( two - way anova ) indicated overall significant differences between the two 5-ht protocols ( f1,18 = 30.12 , p < 0.001 ) .
pair - wise comparisons ( snk tests ) indicated that compared to the standard protocol , the enhanced protocol induced significantly higher levels of creb1 phosphorylation immediately ( q2 = 6.94 , n = 4 , p < 0.001 ) , and 18 h after treatment ( q2 = 3.69 , n = 7 , p < 0.02 ) .
both long - term synaptic plasticity and modifications in creb1 are thought to be key processes underlying long - term memory .
having demonstrated that the enhanced protocol improved ltf and phosphorylation of creb1 , the enhanced protocol was hypothesized to produce stronger learning than the standard protocol in behaving animals .
4 ) . statistical analyses ( two - way anova ) indicated significant differences among the three training groups ( f2,82 = 11.77 , p < 0.001 ) .
pair - wise comparisons ( snk tests ) indicated that both the standard protocol ( q2 = 3.12 , n = 12 , p < 0.04 ) and enhanced protocol ( q3 = 4.13 , n = 14 , p < 0.02 ) induced lts that lasted 1 d. the lts that was induced by the standard protocol , however , did not persist for five days ( q2 = 0.38 , n = 14 ) .
in contrast , the lts that was induced by the enhanced protocol persisted for at least five days ( q3 = 5.48 , n = 9 , p < 0.001 ) .
the results indicated that computational approaches could be used to design a stimulus protocol that increases long - term synaptic plasticity , transcriptional activation , and long - term sensitization .
interestingly , the non - uniform distribution of isis in the enhanced protocol was in marked contrast to the fixed intervals used in previous protocols for behavioral training in aplysia .
for example , a form of training with expanding isis appears to lead to enhanced performance in verbal recall tasks .
the mechanistic underpinnings of this form of training are not well understood , and therefore provide little guidance for designing protocols that maximize retention .
the present results suggested that modeling the dynamics of biochemical cascades could play an essential role in understanding why specific training protocols are more effective than others , and thereby , help guide the development of better learning paradigms .
a goal of the present study was to develop training protocols that enhanced learning and memory . however , the model described in fig .
1a was also compatible with previous empirical results in which a massed training protocol ( 25 min continuous 5-ht ) produced significantly less ltf than a spaced protocol ( five , 5-min pulses of 5-ht ) .
simulations of this massed protocol produced a peak level of inducer of only 0.00016 m , which was at the low end of the distribution illustrated in supplementary fig .
, the model was able to predict submaximal performance , which helped to increase confidence in its predictive power .
these results indicate that the improvement found with the enhanced protocol did not represent a narrow range of model parameter values , but occurred for various combinations of values .
such behavior is expected for robust models of physiological processes , in which optima should not be sharply peaked at very specific parameter values . in humans ,
a number of cognitive models have been proposed to explain the superiority of spaced training or practice , but no consensus has emerged as to their usefulness or generality .
it is likely that these models have had limited predictive ability , at least in part , because they are not based on known neurobiological processes .
in contrast , the present model was based on well characterized biochemical cascades in individual neurons that participate in the induction and expression of ltm .
although the present model was limited to a description of the dynamics of biochemical cascades associated with the initial induction of ltm and to relatively brief training sessions ( e.g. , 1.5 h ) , the model successfully predicted a protocol that enhanced synaptic strength and memory for at least five days after stimulation .
these results confirmed that important aspects of the induction of ltm occur during the initial training phase .
for example , the slow dynamics of additional biochemical cascades contribute to ltm ( e.g. , changes in the synthesis of transcription factors such as creb1 and creb2 ) .
, the model did not include descriptions of postsynaptic protein kinase c , upregulation of local protein synthesis in the vicinity of the synapse or synthesis of the peptide sensorin in response to retrograde signals , which are involved in ltf .
finally , the model did not account for the differences in the time course and magnitude of ltf ( fig .
4 ) . several factors may contribute to this discrepancy , including a nonlinear transformation of ltf into motor neuron activity and plasticity at other loci in the neural circuit that contribute to the behavioral enhancement .
an expanded model that includes the dynamics of additional biochemical cascades and neural circuit elements would presumably have greater predictive capability .
the present results indicated the feasibility of using computational methods to assist in the design of training procedures that enhance learning
. it will be important to determine the extent to which the computational approaches can be applied to other memory systems , and ultimately , applied to training procedures used to improve human cognition .
a detailed description of the methods is available in the online version of the paper at http://www.nature.com / natureneuroscience/. note : supplementary information is available on the nature neuroscience website .
 learning and memory are influenced by the temporal pattern of training stimuli . 
 the mechanisms that determine the effectiveness of a particular training protocol are not well understood , however . 
 the hypothesis that the efficacy of a protocol is determined , in part , by interactions among biochemical cascades that underlie learning and memory was examined . 
 previous studies suggest that the pka and erk cascades are necessary to induce long - term synaptic facilitation ( ltf ) in aplysia , a neuronal correlate of memory 
 . a computational model of the pka and erk cascades was developed , and used the model to identify a novel training protocol that maximized pka / erk interactions . in vitro studies confirmed that the protocol enhanced ltf . 
 moreover , the protocol enhanced levels of phosphorylation of the transcription factor creb1 . behavioral training confirmed that long - term memory also was enhanced by the protocol . 
 these results illustrate the feasibility of using computational models to design training protocols that improve memory . 