ERP Studies Of Structural Processing Of Arithmetic Expressions

Mathematical cognition has been a hot topic in the field of cognitive neuroscience.Traditional research on mathematical cognition mainly focused on numerical cognition and arithmetic calculation.Substantial knowledge has been attained in these aspects.On the other hand,like human language,arithmetic expressions are also cognitive sequences followed certain rules. Hauser et al.(2002) suggested that "recursion" was a marker of high-order mathematical and linguistic abilities in humans.Subsequently,Houd(?) & Tzourio-Mazoyer(2003) claimed that the generative power of grammar might provide a general cognitive template and a specific constitutive mechanism for "syntactic" mathematical operations involving recursiveness and structure dependency.These proposals included a common hypothesis,that there can be common basic neural mechanisms for the processing of both linguistic and mathematical structures.For instance,both the computation of arithmetic expressions involving embedded operation(e.g.,"5×(3+2)") and sentences including embedded clause(e.g.,"The man who killed the lion was angry") need the comprehension of the function role of the embedded structures.The hypothesis that there are common mechanisms for the processing of both linguistic and mathematical structures becomes a new hot theme.However,till now,the empirical evidence for this issue is insufficient and contradictory.Event-related potential(ERP) technique is suitable for investigating the online processing of arithmetic expressions and sentences.A large body of ERP studies on language processing has found the a late positivity named P600,which occurs 300-900 ms after the stimulus onset and distributes in the centroparietal areas on scalp,is related to syntactic processing.It is elicited by syntactic violations or syntactically correct but complex sentences.By contrast,ERP studies on structural processing of arithmetic expressions are sparse. It is unclear which ERP effects are correlated with structural processing of arithmetic expressions. Moreover,the existing few studies made inconsistent conclusions about whether there are common mechanisms of structural processing in language and mathematics.Mart(?)n-Loeches et al.(2006) investigated the ERP correlates of the processing of the "mathematical syntax problems".Given the lack of the early anterior negativity for the three types of mathematical problems and the functionally dissociable LPCs associated with violations of arithmetic structures,the authors argued that the neural mechanisms for structural processing in the domain of language and mathematics are distinct.However,a study by N(?)ez-Pe(?)a & Esceras (2004) made the reversed conclusion.In that study,sequences comprised of seven numbers were used.A centroparietal LPC was evoked by the incorrect ending numbers.Its amplitude was sensitive to the degree of the violation,being larger for the great than small violations.The authors explained the LPC as a parallel of the P600,being an index of structural integration of the arithmetic sequences.The results were taken as evidence of a similar neurophysiological mechanism underlying processing of both arithmetical rules and linguistic syntactic structures.Based on the current research status,the present study investigated the neural correlates of structural processing of arithmetic expressions by using ERPs.Well-formed arithmetic expressions with different complexity were utilized.If a high priority operation follows a low priority operation during the sequence of normal(left-to-right) reading,the former was named as an embedded operation.The structural complexity was defined by the number of the embedded operations included in one expression(zero,one,or two).It is presumed that the structural complexity increases with the number of the embedded operations.The working hypothesis of the thesis is that the structural processing of arithmetic expressions follows the "simplicity principle", as does the syntactic processing of sentences.This principle denotes that the processor always predict or build up the simplest structural relation analysis based on the available information.If the actual stimuli are inconsistent with the simplest analysis,additional processing cost will be elicited and a P600-like effect will be observed.Since to date there is no mature stimuli presentation paradigm,the present thesis tentatively employed the chunk presentation procedure (Experiment A,B and C) and the single character presentation procedure(Experiment D and E). While the former visually presented the expressions chunk-by-chunk(each chunk comprised of a number and an operator;in some cases also a parenthesis),the latter presented the expressions character-by-character.The participants were asked to calculate the expressions according to standard rules.Experiment A aimed at establish the ERP correlates of the structural processing of arithmetic expressions.Four types of expressions were used:AHA(e.g.,"9+5×(3+2)=") was highly complex,AMA(e.g.,"6+2×8-1=") was moderately complex,ANA(e.g.,"5+7-3+2=") and ANM(e.g.,"2×5+4+6=") were simple.The results showed that two LPCs were elicited by AHA2 compared to ANA2,ANM2 and AMA2.The first LPC was within 300-400 ms,and parieto-occipitally distributed.The second LPC was within 400-650 ms,and centroparietally distributed.The two LPCs could be the analogs of the P600,reflecting the mismatch between the actual stimuli and the structural expectations.However,different from the predictions,AMA2 did not elicit LPCs relative to ANM2 and ANA2.This arose two althernative explanations for the LPCs by AHA2.Firstly,they could be the P3b,being related to the less probability of the parenthesis.Secondly,as the parenthesis indexed the double embedded operations,which meant the calcualtion difficulty would be greater,the two LPCs by AHA2 could be related to task difficulty anticipation.The aim of Experiment B was to rule out these two possiblilties.Experiment B used three types of expressions:BdHA(e.g.,"5+4×(6+3)="),BeHA(e.g.,"7+ 2×(1+3)=") and BFA(e.g.,"6+5+(2+7)=").The results showed that after the probability difference of the parenthesis was excluded,BHA2(combiation of BeHA2 and BdHA2) still elicited two LPCs relative to BFA2.This indicated that stimili probability was not the main cause of the LPCs.Moreover,the two LPCs did not differentiate between BeHA2 and BdHA2, indicating task difficulty anticipation was not the source of the LPCs.However,the possiblity that the stimili probability and the strucutral complexity co-influenced the LPCs could not be excluded.Experiemnt C aimed at testing:(1) whether the equal LPCs amplitude for AMA2 and ANA2/ANM2 were due to the calculation inhibition-related negativity during 320-430 ms by AMA2 and the calcualtion-related positivity by ANA2/ANM2,and(2) whether the stimili probability and strucutral complexity co-influenced the LPCs by AHA2.Four types of arithmetic expressions were used:CHA(e.g.,"8+b×(2+6)="),CEA(e.g.,"5+(c×4)+6="),CMA(e.g., "5+(c×4)+6=") and CNA(e.g.,"2+a+9+7=").A P600-like LPC was elcited by CHA2 and CEA2 comapred to CMA2 and CNA2 within 300-530 ms.It suggested that the amplitudes of the LPC associated with the violations of the strucutral expectations were proportional to the structural complexity.Meanwhile,because the LPC did not differentiate between CHA2 and CEA2,it was possible that the LPC elicited by the chunk containing a parenthesis was the composite effects of the strucutral expectation violation and the context updating for the less probable sitmuli.Experiemnt D was set out to study the ERP effects correlated with strucutral processing evoked by the single parenthesis and operator.Single character presentation procedure was used to present five types of arithmetic expressions:DMA(e.g.,"3+4×8="),DMM(e.g.,"3×(4+9) ="),DNA(e.g.,"3+4+6="),DNM(e.g.,"2×6+5=") and DFA(e.g.,"(3+4)×5=").The results showed that a centroparietal LPC within 400-600 ms was elicited by DMM3(the third character of DMM,the same nomenclature for the following text) relative to DNM3.A right parieto-occipital LPC was elicited by the parenthesis occured at the first position relative to the numbers at the first position within 350-550 ms.Profile Analyses(McCarthy & Wood,1985) showed that the two LPCs correlated with the parenthesis were different in topographies, indicating dfferent underlying processes for the two LPCs.While the LPC by DMM3 might reflect the mismatch between the actual stimuli and the strucutral expectations,the LPC by the parenthesis at the first position might reflect the context updating processes of the less probable stimuli.A posterior LPC within 400-600 ms and a central LPC within 600-750 ms were elicited by DMA4 relative to DNM4.Meanwhile,a posterior LPC within 400-600 ms was elicited by DNA4 relative to DNM4 as well.The two LPCs within 400-600 ms did not differentiate in topograhpies. There were two approaches to explain the two LPCs by DMA4.The first was that the two LPCs represented the two stages of strucutral reanalysis processes,i.e.,the diagnosis of the the need of reanalysis and the actual revision of the strucutral relations.The second was that the LPC within 400-600 ms was a P300,associated with task difficulty anticipations,whereas the LPC within 600-750 ms was the genuine reflection of the strucutral reanalysis processes.The aim of Experiemnt E were to clarify the meaning of the LPC by DNA4 and the LPC by DMA4(400-600 ms).Six types of arithmetic expressions were utilized:EMAL(e.g.,"a+5×4 ="),EMAN(e.g.,"3+9×a="),ENAL(e.g.,"b+3+5="),ENAN(e.g.,"3+7+b="),ENML (e.g.,"c×8+7=") and ENMN(e.g.,"4×6+c=").The LPC did not differentiate for ENAN4 and ENAL4,suggesting it was unrelated to the calculation processes.The LPC within 400-600 ms was elicited by EMAL4 compared to ENML4 and ENMN4,suggesting that the LPC by DMA4 within 400-600 ms did not reflect the first stage of the strucutral reanalysis processes.Instead,it was most likely to be a manifestation of the task difficulty anticipations.The present thesis has proved that the P600-like LPC is the ERP correlate of the structural processing of complex arithmetic expressions.Moreover,both the amplitude and latency of the P600-like LPC are sensitive to the strucutral complexity of the arithmetic expressions.The more complex the arithmetic expression,the larger the LPC amplitude,and the later the LPC latency. Previous studies have found that the P600 is related to anomalies of syntax,abstract structure, harmonic structures,and arithmetic expressions,as well as to syntactically correct but complex sentence structures.The present series of experiments provide new evidence for the above line of research,that the P600-like LPCs can also be elicited by the structurally correct but complex arithmetic structures.Taken together,the P600 can be seen as a general index of the strucutral processing for all rule-governed sequences.Both the mismatch between actual input and the strucutral expectation and the structural reanalysis process have been documented in previous language studies(e.g.,Friederici,1998; Osterhout & Holcomb,1992,1994).Importantly,the P600 related to these processes reported in the language studies is comparable to the strucutral LPCs found in the present thesis in both morphological and functional terms.The results of the present thesis indicate that,the structural processing of the arithmetic expressions depends on the comprehension of the structural dependency relations between the operands.The structural dependency relations are based on abstract rules,and may work above the specific symbol systems(numbers or words).The processing of the structural dependency relations between the elements is very likely one of the common mechanisms of the structural processing shared by mathematics and language.The thesis has important theoretical significance.On the one hand,it adds valuable knowledge of the mechanisms of structural processing aspects in the field of mathematical cognition.On the other hand,it provides new empirical evidence for the hot dispute about the simiarity of the structural processing between language and mathematics in the field of cognitive neuroscience.