Fractional Dynamics and Multi-Slide Model of Human Memory Ihor Lubashevsky ([email protected]) University of Aizu, Tsuruga, Ikki-machi, Aizu-wakamatsu, 965-8580 Fukushima, Japan
Bohdan Datsko (b [email protected]) Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences, Naukova st., 3b, Lviv, 79601, Ukraine
Abstract larger composed chunks and, finally, their hierarchical net- work. The notion of chunk is general, therefore, it is rather We propose a single chunk model of long-term memory that problematic to define it more precisely, for discussion and combines the basic features of the ACT-R theory and the mul- tiple trace memory architecture. The pivot point of the devel- history see, e.g., a review by Anderson and Lebiere (1998). oped theory is a mathematical description of the creation of Second, each chunk individually is characterized by its new memory traces caused by learning a certain fragment of strength F which determines also the information retention, information pattern and affected by the fragments of this pat- tern already retained by the current moment of time. Using namely, the probability of successful retrieval of the corre- the available psychological and physiological data these con- sponding information from a given chunk (Anderson, 1983; structions are justified. The final equation governing the learn- Anderson & Schooler, 1991). Since the classical experiments ing and forgetting processes is constructed in the form of the differential equation with Caputo type fractional time deriva- of Ebbinghaus (1885/1913) a rather big data-set about the re- tive. Several characteristic situations of the learning (contin- tention ability of human memory has been accumulated for uous and discontinuous) and forgetting processes are studied time scales from several minutes up to a few weeks. It has numerically. In particular, it is demonstrated that, first, the “learning” and “forgetting” exponents of the corresponding been figured out that the memory strength F decays with time power laws of the memory fractional dynamics should be re- t according to the power-law (Wickelgren, 1972, 1974; An- garded as independent system parameters. Second, as far as derson & Schooler, 1991; Wixted & Ebbesen, 1991, 1997), the spacing effects are concerned, the longer the discontinu- ous learning process, the longer the time interval within which i.e., exhibits the asymptotic behavior a subject remembers the information without its considerable lost. Besides, the latter relationship is a linear proportionality. 1 F(t) ∝ d , (1) Keywords: human memory; memory trace; chunk; forgetting; t learning; practice; spacing effects; power law; fractional dif- ferential equations. where the exponent d is a certain constant. It should be noted that, in general, this dependence meets the second Jost’s law, the increment of the strength decay becomes weaker as time Psychological and Physiological Background goes on (see, e.g., a review by Wixted (2004)). Appealing There are a number of approaches to understanding and de- at least to the data-set collected by Ebbinghaus (1885/1913) scribing processes in human mind. They belong to differ- and analyzed by Anderson and Schooler (1991) as well as one ent levels of abstraction, ranging from neural and biochemi- collected and studied by Wixted and Ebbesen (1991, 1997) cal processes in the brain up to philosophical constructions, the exponent d seems to be rather universal and can be esti- and study its different aspects. In the present work we fo- mated as d ∼ 0.1 − 0.2. cus our attention on the phenomenological (psychological) The third postulate concerns the multiple-trace arrange- description of human memory dealing with it as a whole, i.e., ment of human memory. It assumes that each attempt of without reducing the corresponding mental functions to the learning and memorizing some information fragment pro- real physiological processes implementing them. A review duces a separate trace m in human memory. So the corre- of advances made in this scope during the last decades can sponding cognitive unit, a chunk, is actually a collection {mi} be found in a monograph by Anderson (2007) who inspired of many memory traces and its strength F is the sum of their the development of the ACT-R concept in cognitive science, individual activation levels { fi} (Hintzman, 1986) a modern theory about how human cognition works. The ACT-R theory operates with three types of human F(t) = ∑ fi(t). (2) m ∈Chunk memory, sensory, short-term, and long-term ones (Atkinson i & Shiffrin, 1968) and accepts, in particular, the following ba- Evidence collected currently in physiology (see, e.g., work by sic postulates. Yassa and Reagh (2013) and references therein) partly sup- First, the declarative (long-term) memory is organized in ports this concept. Its implementation in physiological terms chunks (Miller, 1956), certain cognitive units related to some is reduced to the Multiple Trace Theory (MTT) developed by information objects. At the first approximation the learning, Nadel and Moscovitch (1997) appealing to the role of the hip- memorizing, and retrieval of a given object proceeds via the pocampus in the encoding of new memory traces as well as creation and evolution of the corresponding chunk. Naturally, the retrieval of all the previous traces, including remote ones. chunks can interact with one another, in particular, forming The preceding alternative of MTT is the Standard Model of
2597 Systems Consolidation (SMSC: Squire and Alvarez (1995)). It assumes the hippocampus to “teach” the cortex a memory trace strengthening the connectivity between its individual el- ements over time and, finally, consolidating the memory. Recently Yassa and Reagh (2013) have proposed a Com- petitive Trace Theory (CTT) combining elements SMSC and MTT. It suggests that when a memory is reactivated by a new time cue, the hippocampus acts to re-instantiate the original mem- ory traces, recombine their elements in the episodic memory, and add or subtract individual contextual features. As a re- moment of transferring sult, a new memory trace overlapping with the original ones a new pattern fragment is created and ready to be stored in the neocortex. However, from the working memory in contrast to MTT, CTT supposes that the memory traces are to the declarative memory not stored in parallel but compete for representation in the neocortex. Two relative phenomena occur here: consolida- time tion and decontextualization. First, overlapping features in the memories should not compete for representation and thus are strengthened, i.e., consolidated. Second, non-overlapping features should compete with one another resulting in mutual inhibition and, as a result, memories become decontextual- ized. Nadel and Moscovitch (1998) proposed the reactivation of memory traces to strengthen also the links between the traces too. The concept of such a multi-trace consolidation can be regarded as the fourth postulate of the ACT-R theory. As the fifth postulate we note the following. The hip- pocampus is involved in the “reconstruction” rather than the “retrieval” of the memory. So, as CTT states, new memory Figure 1: Illustration of the used model for the chunk evo- traces are only partially but not completely overlap with the lution as new slides are created in order to compensate the original traces. It is due to the hippocampus capability of degradation of the previous ones. supporting rapid encoding of unique experiences by orthogo- nalizing incoming inputs such that their interference is min- imized, which is termed pattern separation; the available ev- the chunk interaction and the memory consolidation are chal- idence for this feature was recently discussed by Yassa and lenging problems on their own and require individual investi- Stark (2011). This pattern separation together with the corre- gation. sponding pattern completion via creating new memory traces endows our episodic memory system with richness, associa- Single Chunk Model tivity, and flexibility (Yassa & Stark, 2011). {m(t0)} Finally, the ACT-R theory accepts an important general- A chunk M is considered to be a collection of traces {t0} ization about expansion (2). It assumes that the individual created in the working memory at time moments and stored in the declarative (long-term) memory. These traces activation levels fi(t) of memory traces decreases with time t also according to the power law and, after formation, their will be also called slides for reasons apparent below. Let us individual dynamics is mutually independent. Thereby the assume that the chunk M as a whole contains a certain frag- strength F(t) of the corresponding chunk is the superposition ment of information, a pattern P, so all its slides retain differ- (Anderson, 1983; Anderson & Schooler, 1991) ent fragments of this pattern. The time evolution of the chunk M as a whole unit is de- C scribed in terms of its strength F(t), the measure of the rel- F(t) = i , (3) ∑ d ative volume of the pattern pieces that are retrievable at the m ∈Chunk (t −ti) i current moment of time t (Anderson, 1983, 2007). The in- 0 where {Ci} are some constants and ti is the time moment dividual evolution of the slides {m(t )} is characterized by 0 when the chunk mi was created. It should be noted that ex- similar quantities { f (t,t )} depending on the current time t 0 pression (3) does not take into account the chunk interaction. and the time t when the corresponding slide was created and Our following constructions will be based on these pos- stored. tulates. In the present work we will confine our considera- tion to the dynamics of a single chunk and ignore the effects Slide Formation Mechanism of memory consolidation which are likely to be crucial only The chunk slides are assumed to be created according to the on relatively large time scales. Mathematical description of following scenario illustrated in Fig. 1. Memory continuously
2598 looses some fragments of the pattern P. So when at the cur- which is related directly to the assumption about the reduction rent moment of time t the chunk M as a whole is retrieved of the slide capacity at the moment of its creation. As time from the declarative memory only some its fragment Pt can goes on, the strength of all the slides decreases and without be retrieved, which is characterized by the value F(t) < 1. addition learning the strength F(t) of the chunk M as whole Then addition practice or learning is necessary to reconstruct is written as the sum of all the slides created previously the initial pattern P. Therefore a new slide m(t) to be created F(t) = C(t0) f (t,t0). (9) during this action has to contain, at least, the fragment P \ Pt . ∑ t0
2599 This expression would lead exactly to formula (5b) if the time 1.0 scale T were independent of F. However, ansatz (5b) has been chosen rather arbitrary keeping in mind only the basic 0.8 features it should possess. So we are free to replace it by the expression stemming from model (15). The last equality 0.6 in model (15) formally coincides with formula (9) except for W0 the fact that the value F(t) has to decrease additionally due 0.4 to time evolution of temporal elements created previously. learning rate, However, all the “microscopic” time scales, in particular, τ0 0.2 attention quantity, W quantity, attention 0 and τm, are related directly to the interval within that a new 0 chunk strength,F 1 slide is created in the working memory and, then, transfered 0.0 to the declarative memory. It enables us to ignore this de- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 crease in the value F(t) during the time interval τ. As a result chunk strength,F expression (12) can be reduced to the following integral
t Figure 2: The accepted ansatz for the dependence of the Z 0 0 −d 0 W(t ) (t −t ) 0 learning rate L(F,W) (in dimensionless units) on the strength F(t) = [1 − F(t )] 0 1 + dt . (16) T[F(t )] τ0 F of a given chunk. The used approximation for the attention −∞ quantity W(F) depending on the chunk strength is shown in Moreover, due to the integral inset. The values of ε = 0.01, α = 0.5, β = 0.1, and Fc = 0.95 Z t 1 were used in constructing the present curve. d dζ 0 ζ In order to avoid some artifacts in numerical simulation we converging at the lower boundary ζ = 0 for the exponent d < will accept an additional assumption that it is not possible to 1 we can replace kernel (12) by the corresponding power-law get strictly the limit value of the chunk strength F = 1 by kernel (t −t0)−d τd learning a subject. Indeed, the closer the chunk strength to 1 + ⇒ 0 . (17) unity, the more attention is necessary for a human to recog- τ (t −t0)d 0 nized which piece of information is unknown for him. As a After this replacement expression (16) reads result, we introduce a certain critical value Fc ' 1 such that, t when the chunk strength F exceeds it, F > Fc, a human con- Z 0 d 0 W(t ) τ0 0 siders the success of learning to be achieved and it attention F(t) = [1 − F(t )] 0 0 d dt . (18a) T[F(t )] (t −t ) to the learned subjects disappears, i.e., W(F) = 0 for F > Fc. −∞ It is illustrated in Fig. 2. or using ansatz (6) Results and Discussion t Z 1 0 α 0 β 0 1 0 The characteristic features of the system dynamics were stud- F(t) = 1−d [ε + F(t )] [1 − F(t )] W(t ) 0 d dt . τm (t −t ) ied numerically using the explicit 2-FLMM algorithm of sec- −∞ (18b) ond order (Galeone & Garrappa, 2009) for solving equa- In deriving expression (18b) we have aggregated the ratio tion (19). Figure 3 presents some of the obtained results. d (τ0/τm) into the quantity W. So, first, the integral equa- Fractional Dynamics of Forgetting tion (18a) contains only one microscopic time scale τ re- m Plot I (Fig. 3) shows the forgetting dynamics under the “ba- garded as a certain model parameter. Second, the dimension- sic” conditions matching the following hypothetical situation. less quantity W(t) describes the attention to the subject during At the initial moment of time t = 0, a subject starts to learn the learning process. If W = 1 then the given pattern can be P continuously an unknown for him information pattern being learned completely for a time interval about τ. retained in a single chunk and at time t = T ends this process Finalizing the given construction we will assume that the L when the chunk strength gets its limit value F ' 1. As time learning process was initiated at time t = 0 and before it no c goes on, the chunk strength F(t) decreases, which specifies information about the pattern was available, i.e., for t < 0 P the decay of retrievable information. As should be expected, the value F(t) = 0. In this case the integral equation (18b) can the asymptotics of F(t) is of the power law and looks like a be rewritten in the form of the following differential equation straight line on the log-log scale plot. Naturally, in a certain with time fractional derivative of the Caputo type neighborhood QL of the time moment t = TL this asymptotics (1−d) C (1−d) g γ does not hold. However, for small values of the exponent d τm · D F = (ε + F) (1 − F) W(t). (19) (for d = 0.2 in Plot I) this neighborhood is narrow and be- It is the desired governing equation for learning and forgetting comes actually invisible in approximating experimental data processes. even with weak scattering.
2600 Figure 3: Some results of numerical simulation. First, it is the dynamics of forgetting (I) and learning (II) under the “basic” conditions when a subject learns an unknown information continuously until he gets the local (temporal) success in time TL. Second, it is the spacing effects in discontinuous learning (IIIa) and the following forgetting (IIIb). In this case the subject learns an unknown information via a sequence of trials of a fixed duration until he gets the final success at a certain moment of time TL. Naturally, the longer the time spacing between two successive trials, the larger the number NL of trials and the longer the total time interval TL are necessary for this. In simulations the following parameters ε = 0.01, α = 0.5, β = 0.1, Fc = 0.95, dt = 0.001, and TL|d=0.2 = 1.70, TL|d=0.5 = 1.52 were used as a common setup, the other individual values are shown in the corresponding plots.
Fractional Dynamics of Continuous Learning determine the set of quantities required for characterizing hu- man long-term memory, the “forgetting” and “learning” ex- Plot II exhibits the learning dynamics under the same “basic” ponents, d and dL, may be regarded as independent parame- conditions. The growth of the chunk strength F(t) is visual- ters. ized again in the log-log scale for various values of the pa- rameters determining how the learning rate L(F,W) changes Spacing effects during the process (they are given in the inset). As seen, the function F(t) strictly is not of the power law. However, if it is Plots IIIa and IIIb illustrate the found results in the case mim- reconstructed from some set of scattered experimental points icking the discontinuous learning process. It again assumes a as the best approximation within a certain class of functions, subject to start learning an initially unknown information pat- a power law fit (linear ansatz in the log-log scale) may be ac- tern being retained in one chunk during the process, however, cepted as a relevant model. It allows us to introduce an effec- now he does not do this continuously. Instead, the learning d tive exponent dL of the approximation F(t) ∝ t L . Appealing proceeds via a sequence of trials of a fixed duration that are again to Plot II, we draw a conclusion that this effective ex- separated by some time gap (spacing) until the subject gets ponent depends not only on the “forgetting” exponent d but the success at a certain time moment TL. Naturally, the longer also on the other system parameters. Thereby, in trying to the spacing, the longer the total time TL as well as the larger
2601 the number NL of trials required for this. So, in order to com- Cepeda, N., Coburn, N., Rohrer, D., Wixted, J., Mozer, M., pare their characteristic properties let us renormalize the time & Pashler, H. (2009). Optimizing distributed practice. scale in such a way that the learning process end at t = 1 in Experimental Psychology (formerly Zeitschrift fur¨ Exper- dimensionless units, in other words, the time is measured in imentelle Psychologie), 56(4), 236–246. units of TL. In this case, as seen in Plot IIIa and IIIb, the main Cepeda, N., Pashler, H., Vul, E., Wixted, J., & Rohrer, D. characteristics of the shown processes become rather similar (2006). Distributed practice in verbal recall tasks: A review with respect to the dynamics of learning and forgetting. This and quantitative synthesis. Psychological Bulletin, 132(3), result poses a question about optimizing a learning process 354–380. by dividing it into rather short trials separated by relatively Ebbinghaus, H. (1885/1913). Memory: A Contribution to long time intervals. This effect is also called the distributed Experimental Psychology. New York: Teachers College, practice, an analysis of available experimental data can found, Columbia University. (translated to English by Henry Al- e.g., in review by Cepeda, Pashler, Vul, Wixted, and Rohrer ford Ruger and Clara E. Bussenius) (2006) and Cepeda et al. (2009) as well. At least, within the Galeone, L., & Garrappa, R. (2009). Explicit methods for framework of the present fair simple model an increase of fractional differential equations and their stability proper- the time spacing gives rise, on one hand, to the growth of ties. Journal of Computational and Applied Mathematics, the learning duration but, on the other hand, enables one to 228(2), 548–560. remember this information for a longer time without its con- Hintzman, D. (1986). “Schema Abstraction” in a Multiple- siderable lost. Trace Memory Model. Psychological Review, 93(4), 411– 428. General Comments Miller, G. (1956). The magical number seven, plus or minus As far as the theoretical aspects of the present research are two: some limits on our capacity for processing informa- concerned, we note appealing to the obtained results that the tion. Psychological Review, 63(2), 81–97. multiple trace concept of memory architecture requires an in- Nadel, L., & Moscovitch, M. (1997). Memory consolidation, dividual mathematical formalism irreducible to the classical retrograde amnesia and the hippocampal complex. Current notions created in physics. In particular, even at the “micro- opinion in neurobiology, 7(2), 217–227. scopic” level dealing with slides (traces) the system dynamics Nadel, L., & Moscovitch, M. (1998). Hippocampal contri- is not reduced to the motion in a certain phase space but con- butions to cortical plasticity. Neuropharmacology, 37(4), tinuous generation of such phase spaces and their interaction 431–439. with one another become the key elements of the correspond- Squire, L., & Alvarez, P. (1995). Retrograde amnesia and ing theory. memory consolidation: a neurobiological perspective. Cur- Besides, the governing equation (19) admits the following rent Opinion in Neurobiology, 5(2), 169–177. interpretation. Its left-hand side describes “internal” evolu- Wickelgren, W. (1972). Trace resistance and the decay of tion of human memory on its own, whereas the right-hand long-term memory. Journal of Mathematical Psychology, side plays the role of “sources” generating new elements of 9(4), 418–455. memory. This approach can enhance the development of hu- Wickelgren, W. (1974). Single-trace fragility theory of mem- man memory theory by singling out the phenomena to be ad- ory dynamics. Memory & Cognition, 2(4), 775–780. dressed. Wixted, J. (2004). 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