Circadian timing is generated through a unique series of autoregulatory interactions termed the molecular clock. Behavioral rhythms subject to the molecular clock are well characterized. We demonstrate a role for Bmal1 and Clock in the regulation of glucose homeostasis. Inactivation of the known clock components Bmal1 (Mop3) and Clock suppress the diurnal variation in glucose and triglycerides. Gluconeogenesis is abolished by deletion of Bmal1 and is depressed in Clock mutants, but the counterregulatory response of corticosterone and glucagon to insulin-induced hypoglycaemia is retained. Furthermore, a high-fat diet modulates carbohydrate metabolism by amplifying circadian variation in glucose tolerance and insulin sensitivity, and mutation of Clock restores the chow-fed phenotype.  Bmal1  and  Clock,  genes that function in the core molecular clock, exert profound control over recovery from insulin-induced hypoglycaemia. Furthermore, asynchronous dietary cues may modify glucose homeostasis via their interactions with peripheral molecular clocks.Bmal1Clock,

Introduction The master clock, which, in mammals, resides in the hypothalamic suprachiasmatic nucleus (SCN), is thought to synchronize multiple peripheral oscillators to ensure temporal coordination of behavior and metabolism. Peripheral clocks amplify or dampen central rhythms or exhibit autonomous behavior to facilitate local adaptive responses ( Hastings et al. 2003 ). The central clock may communicate to modulate or entrain rhythms in the periphery via hormones ( McNamara et al. 2001 ) or hemodynamic cues. Asynchronous environmental cues, such as eating, also influence the autonomous behavior of peripheral clocks ( Damiola et al. 2000 ;  Stokkan et al. 2001 ). The variation in sleep and wakefulness (activity) is perhaps the most well-known circadian rhythm. Surgical ablation of the SCN in mice ( Ibuka et al. 1980 ;  Welsh et al. 1988 ) and rats ( Ibuka et al. 1977 ;  Mosko and Moore 1979 ) abolishes the nocturnal burst in locomotor activity. Similarly, disruption and/or mutation of Bmal1 ( Bunger et al. 2000 ) or Clock ( Vitaterna et al. 1994 ), transcription factors that compose the positive limb of an autoregulatory feedback loop in the core molecular clock ( Young and Kay 2001 ;  Reppert and Weaver 2002 ), also impairs circadian behavior. Bmal1 and Clock may influence behavioral rhythms by regulating the firing rate of SCN neurons ( Herzog et al. 1998 ;  Deboer et al. 2003 ). Genes relevant to the molecular clock are also expressed in peripheral tissues ( Akhtar et al. 2002 ;  Kita et al. 2002 ;  Panda et al. 2002 ;  Storch et al. 2002 ;  Oishi et al. 2003 ) where approximately 5%–10% of the transcriptome is subject to circadian oscillation ( Albrecht and Eichele 2003 ). Although the precise role of peripheral clocks and the mechanisms that link them to the SCN remain largely obscure, genetic mutation or deletion has implicated peripheral clocks in the regulation of some aspects of cellular function, including division ( Matsuo et al. 2003 ), estrous cyclicity ( Miller et al. 2004 ), and phospholipid metabolism ( Marquez et al. 2004 ). Glucose and lipid homeostasis are also known to exhibit circadian variation ( Seaman et al. 1965 ;  Malherbe et al. 1969 ;  Gagliardino and Hernandez 1971 ;  Schlierf and Dorow 1973 ). Surgical ablation of the SCN impairs the control of glucose homeostasis ( la Fleur et al. 2001 ). However, the proximity of satiety centres to the SCN has potentially confounded interpretation of these results. Indeed, there is no direct evidence implicating the molecular clock in the regulation of glycaemia or insulin sensitivity (S i ). Our studies revealed a profound role for core clock genes— Bmal1  and  Clock —in regulating recovery from insulin-induced hypoglycaemia. Furthermore, the impact of a high-fat diet (HF) was to amplify the diurnal variation in glucose tolerance and S i  in a manner dependent on the  Clock  gene. These studies suggest that the temporal distribution of a caloric load may influence the response to insulin and that circadian variability in glucose homeostasis may be subject to modulation by asynchronous dietary cues.
Introduction
The master clock, which, in mammals, resides in the hypothalamic suprachiasmatic nucleus (SCN), is thought to synchronize multiple peripheral oscillators to ensure temporal coordination of behavior and metabolism. Peripheral clocks amplify or dampen central rhythms or exhibit autonomous behavior to facilitate local adaptive responses ( Hastings et al. 2003 ). The central clock may communicate to modulate or entrain rhythms in the periphery via hormones ( McNamara et al. 2001 ) or hemodynamic cues. Asynchronous environmental cues, such as eating, also influence the autonomous behavior of peripheral clocks ( Damiola et al. 2000 ;  Stokkan et al. 2001 ).The variation in sleep and wakefulness (activity) is perhaps the most well-known circadian rhythm. Surgical ablation of the SCN in mice ( Ibuka et al. 1980 ;  Welsh et al. 1988 ) and rats ( Ibuka et al. 1977 ;  Mosko and Moore 1979 ) abolishes the nocturnal burst in locomotor activity. Similarly, disruption and/or mutation of Bmal1 ( Bunger et al. 2000 ) or Clock ( Vitaterna et al. 1994 ), transcription factors that compose the positive limb of an autoregulatory feedback loop in the core molecular clock ( Young and Kay 2001 ;  Reppert and Weaver 2002 ), also impairs circadian behavior. Bmal1 and Clock may influence behavioral rhythms by regulating the firing rate of SCN neurons ( Herzog et al. 1998 ;  Deboer et al. 2003 ).Genes relevant to the molecular clock are also expressed in peripheral tissues ( Akhtar et al. 2002 ;  Kita et al. 2002 ;  Panda et al. 2002 ;  Storch et al. 2002 ;  Oishi et al. 2003 ) where approximately 5%–10% of the transcriptome is subject to circadian oscillation ( Albrecht and Eichele 2003 ). Although the precise role of peripheral clocks and the mechanisms that link them to the SCN remain largely obscure, genetic mutation or deletion has implicated peripheral clocks in the regulation of some aspects of cellular function, including division ( Matsuo et al. 2003 ), estrous cyclicity ( Miller et al. 2004 ), and phospholipid metabolism ( Marquez et al. 2004 ). Glucose and lipid homeostasis are also known to exhibit circadian variation ( Seaman et al. 1965 ;  Malherbe et al. 1969 ;  Gagliardino and Hernandez 1971 ;  Schlierf and Dorow 1973 ). Surgical ablation of the SCN impairs the control of glucose homeostasis ( la Fleur et al. 2001 ). However, the proximity of satiety centres to the SCN has potentially confounded interpretation of these results. Indeed, there is no direct evidence implicating the molecular clock in the regulation of glycaemia or insulin sensitivity (S i ).Our studies revealed a profound role for core clock genes— Bmal1  and  Clock —in regulating recovery from insulin-induced hypoglycaemia. Furthermore, the impact of a high-fat diet (HF) was to amplify the diurnal variation in glucose tolerance and S i  in a manner dependent on the  Clock  gene. These studies suggest that the temporal distribution of a caloric load may influence the response to insulin and that circadian variability in glucose homeostasis may be subject to modulation by asynchronous dietary cues.Bmal1ClockClockResults We examined the role of the molecular clock in glucose homeostasis by using mice in which core clock genes are impaired  (Clock mut )  or deficient  (Bmal1 −/− ) . Both plasma glucose and triglycerides were subject to circadian variation in wild-type (WT) mice, peaking at approximately circadian time point 4 (CT4) and CT28 (where CT0 is subjective day beginning at 7 AM, and CT12 is subjective night beginning at 7 PM) ( Figure 1 A and  1 B), as reported previously ( Seaman et al. 1965 ;  Schlierf and Dorow 1973 ). We also observed that corticosterone ( Figure 1 C), which stimulates gluconeogenesis during hypoglycaemia ( Cryer 1993 ), and adiponectin ( Figure 1 D), which has been associated with insulin resistance ( Yamauchi et al. 2001 ;  Maeda et al. 2002 ), oscillated significantly, but out of phase with the glucose and triglyceride rhythms. Diurnal variation in glucose and triglycerides, but not in corticosterone, was disrupted in the mutant mice ( Table 1 ). Figure 1 Circadian Variation of Glucose, Triglyceride, and Hormone Levels in Circulating Blood Plasma from whole blood isolated from unchallenged WT mice at different CTs was analyzed for glucose (A), triglyceride (B), corticosterone (C), and adiponectin levels (D) ( n  = 12 per time point). Results for  Bmal1
 −/−  and  Clock mut  mice are shown in  Table 1 . Table 1 Clock-Controlled Metabolic Rhythms Utilizing peak and trough CTs described in Figure 2A, WT mice showed circadian variation in glucose and triglycerides at CT5 and CT17 that was absent in  Bmal1 −/−  mice, though corticosterone rhythms were preserved in the mice with disrupted circadian rhythms. Values are presented as mean ± SEM;  n  = 7–12 * p  < 0.05 versus corresponding time point (one-way ANOVA with the Kruskal-Wallis test) ND, no data Although there was no clear rhythm in the hypoglyacemic response to insulin, recovery of blood glucose exhibited a robust circadian variation ( Figure 2 A), with an excessive rebound from the effects of insulin evident at subjective dawn (CT19 and CT25) ( Figure 2 A). Insulin caused a profound hypoglyacemic response, independent of clock time, in both  Bmal1
 −/−  and  Clock mut  mice ( Figure 2 B). This response was more pronounced in the former, consistent with the comparative severity of the molecular and behavioral phenotypes between the  Bmal1 −/−  and  Clock mut  animals ( King et al. 1997 ;  Bunger et al. 2000 ). Despite exacerbation of the hypoglycaemic response to insulin in the mutants, the counterregulatory responses of both corticosterone and glucagon were retained ( Figure 2 C and  2 D). Figure 2 Disruption of Genes in the Core Molecular Clock Alters the Response to Insulin (A) Insulin tolerance (IT) was examined in WT mice on CT7, CT13, CT19, and CT25 at 30 min, 60 min, and 90 min after insulin injection ( n  = 12 per time point, * p  < 0.01). (B) IT was examined in WT (black line),  Bmal1
 −/−  (blue line), and  Clock mut  mice (green line) at CT1 (i) and CT13 (ii) ( n  = 6–10, * p  < 0.05, † p  < 0.01, †† p  < 0.001). (C and D) Plasma levels of the counterregulatory hormones corticosterone (C) and glucagon (D) were assessed 60 min after insulin injection in  Bmal1
 −/−  and  Clock mut  mice ( n  = 7, corticosterone assay; samples were pooled for glucagon assay, * p  < 0.05). Gluconeogenesis also contributes to restoration of blood glucose after insulin-induced hypoglycaemia. Consistent with this observation, conversion of exogenously administered pyruvate to glucose, which reflects gluconeogenesis ( Miyake et al. 2002 ), was impaired in the  Clock mut  animals. This impairment was most marked in  Bmal1 −/−  mice, while  Bmal1 +/−  and  Clock mut  mice exhibited an intermediate phenotype when compared with WT littermate controls ( Figure 3 A). Furthermore, activity of the key rate-limiting enzyme of gluconeogenesis, phosphoenolpyruvate carboxykinase (PEPCK), exhibited diurnal variation in the liver and aorta that was blunted in  Clock mut  mice ( Figure 3 B). PEPCK activity in kidney was antiphasic to the rhythm in aorta and liver and was unimpaired in  Clock mut  mice ( Figure 3 B), suggesting tissue-specific regulation of enzyme activity. Figure 3 Impaired Gluconeogenesis in Mice with a Disrupted Circadian Clock (A) Pyruvate tolerance was compared among WT (black line),  Bmal1 +/−  (blue line), and  Bmal1 −/−  (purple line), and  Clock mut  (green line) mice at CT7 ( n  = 6–10). (B) Relative PEPCK activity (units are expressed as luciferase activity × 10 3 ) was measured in liver (i), aorta (ii), and kidney (iii) from WT (white bars) and  Clock mut  mice (green bars). The frequent sampling intravenous glucose tolerance test (FSIGT) was performed to assess more precisely the impact of the molecular clock on sensitivity to insulin. This test provides an estimate of S i , consistent with that obtained by the euglycaemic clamp ( Pacini et al. 2001 ). Additionally, data modeling provides estimates of glucose-mediated glucose disposal (S g ), insulin secretion, and S i . S i  and insulin secretion, but not S g , exhibited a diurnal variation in WT mice fed a regular chow diet (RC) ( Table 2 ). Circadian variation of glucose and lipid homeostasis might condition the metabolic response to asynchronous environmental cues, such as diet, that impinge on S i . Dyslipidemia coincides with insulin resistance in the metabolic syndrome ( Brotman and Girod 2002 ), and a diet high in fat impairs S i  ( Grundleger and Thenen 1982 ;  Coulston et al. 1983 ). Both HF-fed WT and HF-fed  Clock mut  mice increased body weight significantly and to a similar degree in comparison to their age-matched, RC-fed controls ( Table 3 ). Body fat composition averaged 17.6% of lean body mass in RC-fed WT mice, rising to 27.7% ( p  < 0.002) on high-fat feeding. Again, fat composition was not significantly altered by the presence of the  Clock  mutation ( Table 3 ). Table 2 Indices of Noninsulin- and Insulin-Mediated Parameters of Glucose Disposal Derived from Modelling of FSIGT Data at CT1 and CT13 Values are presented as mean ± fractional standard deviation ( n  = 6–7 per group)
 a
 p  < 0.001 versus HF group (two-tailed, Student's t-test)
 b
 p  < 0.001 versus respective group at CT1 G 0 , blood glucose concentration at 1 minute post-glucose bolus; ND, valid S i  value could not be determined Table 3 Body Mass Index Was Assessed in Mice Feeding on RC and HF Values are presented as mean ± SEM;  n  = 13 per group for WT;  n  = 7 per group for subsequent parameters * p  < 0.05 compared to RC (two-tailed Student's t-test); ** p  < 0.01 (two tailed Student's t-test) BMC, bone mineral content Glucose tolerance on an RC trended towards an intolerant phenotype at CT1 versus CT13, but this difference did not attain significance ( Figure 4 A). This is consistent with the temporal variation in insulin secretion observed in RC-fed WT mice in the FSIGT experiment ( Table 2 ). However, when the mice were fed HF for 2 mo, this glycaemic excursion at CT1 evoked by the environmental challenge was amplified and significant (two way analysis of variance [ANOVA];  F  = 63.2,  p  < 0.001) ( Figure 4 A). Similarly, although the hypoglycaemic response to insulin was not different in mice fed regular chow at CT1 and CT13 ( Figure 4 B), the HF induced a significant temporal variation ( Figure 4 B). Thus, the impact of a high fat intake on carbohydrate metabolism in WTs includes an amplification of the diurnal variation in the response to both glucose and insulin. This coincides with a modest impairment in the ability to restore euglycaemia after insulin. A similar impairment resulting from a defect in gluconeogenesis has been observed in rats ( Oakes et al. 1997 ). The mutants failed to exhibit a significant time-dependent variation in their response to glucose or insulin, again reminiscent of the RC-fed, WT phenotype ( Figure 4 ). Figure 4 The Molecular Clock Conditions HF-Induced Circadian Variation in Glucose Homeostasis (A) Glucose tolerance (GT) in RC-fed WT mice (i), HF-fed WT mice (ii), and HF-fed  Clock mut  mice (iii). (B) IT in RC-fed WT mice (i), HF-fed WT mice (ii), and HF-fed  Clock mut  mice (iii) at respective times ( n  = 6–8; * p  < 0.05, † p  < 0.01). Mice were also subjected to extended high-fat feeding (11 mo versus 2 mo). Long-term hyperlipidemia is known to induce frank diabetes with impaired release of insulin ( Johnson et al. 1990 ), in contrast to short-term, high-fat feeding, which increases release of insulin, but impairs the response to it ( Linn et al. 1999 ). The extended HF impaired insulin secretion, reflected by its marked reduction to negative values in HF-fed WT mice ( Table 2 ). The lack of insulin secretion resulted in calculated values of S i  that were imperceptibly low (see Material and Methods). However, S g , insulin secretion, and S i  were restored to a WT phenotype in  Clock mut  mice that were also HF-fed for 11 mo ( Table 2 ). This is, again, consistent with a role for the molecular clock in conditioning the response of glucose metabolism to the intake of dietary fat. Remarkably, mutation of the molecular clock protected against the development of frank diabetes caused by chronic high-fat feeding.
Results
We examined the role of the molecular clock in glucose homeostasis by using mice in which core clock genes are impaired  (Clock mut )  or deficient  (Bmal1 −/− ) . Both plasma glucose and triglycerides were subject to circadian variation in wild-type (WT) mice, peaking at approximately circadian time point 4 (CT4) and CT28 (where CT0 is subjective day beginning at 7 AM, and CT12 is subjective night beginning at 7 PM) ( Figure 1 A and  1 B), as reported previously ( Seaman et al. 1965 ;  Schlierf and Dorow 1973 ). We also observed that corticosterone ( Figure 1 C), which stimulates gluconeogenesis during hypoglycaemia ( Cryer 1993 ), and adiponectin ( Figure 1 D), which has been associated with insulin resistance ( Yamauchi et al. 2001 ;  Maeda et al. 2002 ), oscillated significantly, but out of phase with the glucose and triglyceride rhythms. Diurnal variation in glucose and triglycerides, but not in corticosterone, was disrupted in the mutant mice ( Table 1 ).(Clock mut )(Bmal1 −/− )Clock-Controlled Metabolic Rhythms
Utilizing peak and trough CTs described in Figure 2A, WT mice showed circadian variation in glucose and triglycerides at CT5 and CT17 that was absent in  Bmal1 −/−  mice, though corticosterone rhythms were preserved in the mice with disrupted circadian rhythms. Values are presented as mean ± SEM;  n  = 7–12Bmal1 −/−n* p  < 0.05 versus corresponding time point (one-way ANOVA with the Kruskal-Wallis test)pND, no dataAlthough there was no clear rhythm in the hypoglyacemic response to insulin, recovery of blood glucose exhibited a robust circadian variation ( Figure 2 A), with an excessive rebound from the effects of insulin evident at subjective dawn (CT19 and CT25) ( Figure 2 A). Insulin caused a profound hypoglyacemic response, independent of clock time, in both  Bmal1
 −/−  and  Clock mut  mice ( Figure 2 B). This response was more pronounced in the former, consistent with the comparative severity of the molecular and behavioral phenotypes between the  Bmal1 −/−  and  Clock mut  animals ( King et al. 1997 ;  Bunger et al. 2000 ). Despite exacerbation of the hypoglycaemic response to insulin in the mutants, the counterregulatory responses of both corticosterone and glucagon were retained ( Figure 2 C and  2 D).Bmal1Clock mutBmal1 −/−Clock mutGluconeogenesis also contributes to restoration of blood glucose after insulin-induced hypoglycaemia. Consistent with this observation, conversion of exogenously administered pyruvate to glucose, which reflects gluconeogenesis ( Miyake et al. 2002 ), was impaired in the  Clock mut  animals. This impairment was most marked in  Bmal1 −/−  mice, while  Bmal1 +/−  and  Clock mut  mice exhibited an intermediate phenotype when compared with WT littermate controls ( Figure 3 A). Furthermore, activity of the key rate-limiting enzyme of gluconeogenesis, phosphoenolpyruvate carboxykinase (PEPCK), exhibited diurnal variation in the liver and aorta that was blunted in  Clock mut  mice ( Figure 3 B). PEPCK activity in kidney was antiphasic to the rhythm in aorta and liver and was unimpaired in  Clock mut  mice ( Figure 3 B), suggesting tissue-specific regulation of enzyme activity.Clock mutBmal1 −/−Bmal1 +/−Clock mutClock mutClock mutThe frequent sampling intravenous glucose tolerance test (FSIGT) was performed to assess more precisely the impact of the molecular clock on sensitivity to insulin. This test provides an estimate of S i , consistent with that obtained by the euglycaemic clamp ( Pacini et al. 2001 ). Additionally, data modeling provides estimates of glucose-mediated glucose disposal (S g ), insulin secretion, and S i . S i  and insulin secretion, but not S g , exhibited a diurnal variation in WT mice fed a regular chow diet (RC) ( Table 2 ). Circadian variation of glucose and lipid homeostasis might condition the metabolic response to asynchronous environmental cues, such as diet, that impinge on S i . Dyslipidemia coincides with insulin resistance in the metabolic syndrome ( Brotman and Girod 2002 ), and a diet high in fat impairs S i  ( Grundleger and Thenen 1982 ;  Coulston et al. 1983 ). Both HF-fed WT and HF-fed  Clock mut  mice increased body weight significantly and to a similar degree in comparison to their age-matched, RC-fed controls ( Table 3 ). Body fat composition averaged 17.6% of lean body mass in RC-fed WT mice, rising to 27.7% ( p  < 0.002) on high-fat feeding. Again, fat composition was not significantly altered by the presence of the  Clock  mutation ( Table 3 ).Clock mutpClockIndices of Noninsulin- and Insulin-Mediated Parameters of Glucose Disposal Derived from Modelling of FSIGT Data at CT1 and CT13
Values are presented as mean ± fractional standard deviation ( n  = 6–7 per group)na
 p  < 0.001 versus HF group (two-tailed, Student's t-test)pb
 p  < 0.001 versus respective group at CT1pG 0 , blood glucose concentration at 1 minute post-glucose bolus; ND, valid S i  value could not be determinedBody Mass Index Was Assessed in Mice Feeding on RC and HF
Values are presented as mean ± SEM;  n  = 13 per group for WT;  n  = 7 per group for subsequent parametersnn* p  < 0.05 compared to RC (two-tailed Student's t-test); ** p  < 0.01 (two tailed Student's t-test)ppBMC, bone mineral contentGlucose tolerance on an RC trended towards an intolerant phenotype at CT1 versus CT13, but this difference did not attain significance ( Figure 4 A). This is consistent with the temporal variation in insulin secretion observed in RC-fed WT mice in the FSIGT experiment ( Table 2 ). However, when the mice were fed HF for 2 mo, this glycaemic excursion at CT1 evoked by the environmental challenge was amplified and significant (two way analysis of variance [ANOVA];  F  = 63.2,  p  < 0.001) ( Figure 4 A). Similarly, although the hypoglycaemic response to insulin was not different in mice fed regular chow at CT1 and CT13 ( Figure 4 B), the HF induced a significant temporal variation ( Figure 4 B). Thus, the impact of a high fat intake on carbohydrate metabolism in WTs includes an amplification of the diurnal variation in the response to both glucose and insulin. This coincides with a modest impairment in the ability to restore euglycaemia after insulin. A similar impairment resulting from a defect in gluconeogenesis has been observed in rats ( Oakes et al. 1997 ). The mutants failed to exhibit a significant time-dependent variation in their response to glucose or insulin, again reminiscent of the RC-fed, WT phenotype ( Figure 4 ).FpMice were also subjected to extended high-fat feeding (11 mo versus 2 mo). Long-term hyperlipidemia is known to induce frank diabetes with impaired release of insulin ( Johnson et al. 1990 ), in contrast to short-term, high-fat feeding, which increases release of insulin, but impairs the response to it ( Linn et al. 1999 ). The extended HF impaired insulin secretion, reflected by its marked reduction to negative values in HF-fed WT mice ( Table 2 ). The lack of insulin secretion resulted in calculated values of S i  that were imperceptibly low (see Material and Methods). However, S g , insulin secretion, and S i  were restored to a WT phenotype in  Clock mut  mice that were also HF-fed for 11 mo ( Table 2 ). This is, again, consistent with a role for the molecular clock in conditioning the response of glucose metabolism to the intake of dietary fat. Remarkably, mutation of the molecular clock protected against the development of frank diabetes caused by chronic high-fat feeding.Clock mutDiscussion Maintenance of blood glucose levels within a narrow range is critical to mammalian survival, and environmental cues can trigger appropriate tissue disposition of glucose through adaptive behaviors such as in hibernation ( Castex and Hoo-Paris 1987 ) or the “fight-or-flight” response ( Surwit et al. 1992 ). In this sense, glucose regulation is a fundamental and ancestral defence mechanism. Our studies suggest that Bmal1 and Clock, core components of the molecular clock, contribute substantially to regulation of recovery from the hypoglycaemic response to insulin. However, other mechanisms also impinge on this ancient adaptive response. Thus, impaired recovery from insulin-induced hypoglycaemia is observed in mice lacking proopiomelanocortin. These animals lack adrenal glands and melanocortins and exhibit a defective glucagon response to insulin-induced hypoglycaemia ( Hochgeschwender et al. 2003 ). They contrast with the  Bmal1
 −/−  and  Clock mut  mice, where the counterregulatory hormone response is unimpaired. Thus, steroids, epinephrine, and glucagon appear to facilitate recovery from insulin-induced hypoglycaemia in a manner distinct from, but complementary to, the molecular clock. An assumption intrinsic to our studies is that the phenotypes revealed in the mutant mice are attributable to their function as core elements of the molecular clock. However, as trans-activators, both Clock and Bmal1 may have pleiotropic effects independent of the circadian clock that could impinge on metabolism. Several lines of evidence argue against this hypothesis. First, genes relevant to these metabolic phenotypes display circadian oscillations in their steady-state mRNA levels ( Young et al. 2001 ;  Oishi et al. 2003 ). In addition, the mRNA levels of many of these key proteins are phase-aligned with  Per1  (e.g.,  Enolase 3, Pgam, Transketolase, Lipase, Lpl, Dgat1, Ppar alpha ) or  Per2  (e.g.,  Mod1, Lpl, Pepck, lipin 1 ) (unpublished data). In addition, many of their mRNAs are at lower levels in  Bmal1
 −/−  (e.g.,  Mod1, Pepck, Enolase3, Pgam ) (unpublished data), consistent with a direct role of Bmal1 in their transcription. Thirdly, these metabolic parameters are disrupted in both circadian mutants with the same rank order of potency as the locomotor activity phenotypes  (Bmal1 −/−  > Clock mut ) . Thus, the most parsimonious interpretation is that the observed metabolic deficiencies in the  Bmal1
 −/−  and  Clock mut  mice are due to their roles in the circadian clock, rather than to “off-clock” effects. We observed that the impact of HF on glucose homeostasis was apparently to emphasize the role of the molecular clock. Diet has previously been shown to interact with peripheral clocks. Changes in feeding shift the circadian pattern of gene expression in the liver, but not in the master clock in the SCN ( Damiola et al. 2000 ), demonstrating the importance of food as a cue to circadian control. Individual constituents of food could also provide discrete stimuli. For example, glucose alone can induce rhythmic gene expression in isolated fibroblasts ( Hirota et al. 2002 ). Thus, dietary composition, the size and timing of a feed might all be expected to interact differentially with an underlying circadian regulation of metabolic control. Alterations in dietary content, the availability of “fast food,” inactivity, and sociocultural factors have all been implicated in the emergence of the metabolic syndrome as a major challenge to the public health ( Zimmet et al. 2001 ). However, while mechanistic integration of the diverse elements of the syndrome has proven elusive, our studies suggest that timing may influence the functional consequences of ingesting a caloric load.
Discussion
Maintenance of blood glucose levels within a narrow range is critical to mammalian survival, and environmental cues can trigger appropriate tissue disposition of glucose through adaptive behaviors such as in hibernation ( Castex and Hoo-Paris 1987 ) or the “fight-or-flight” response ( Surwit et al. 1992 ). In this sense, glucose regulation is a fundamental and ancestral defence mechanism. Our studies suggest that Bmal1 and Clock, core components of the molecular clock, contribute substantially to regulation of recovery from the hypoglycaemic response to insulin. However, other mechanisms also impinge on this ancient adaptive response. Thus, impaired recovery from insulin-induced hypoglycaemia is observed in mice lacking proopiomelanocortin. These animals lack adrenal glands and melanocortins and exhibit a defective glucagon response to insulin-induced hypoglycaemia ( Hochgeschwender et al. 2003 ). They contrast with the  Bmal1
 −/−  and  Clock mut  mice, where the counterregulatory hormone response is unimpaired. Thus, steroids, epinephrine, and glucagon appear to facilitate recovery from insulin-induced hypoglycaemia in a manner distinct from, but complementary to, the molecular clock.Bmal1Clock mutAn assumption intrinsic to our studies is that the phenotypes revealed in the mutant mice are attributable to their function as core elements of the molecular clock. However, as trans-activators, both Clock and Bmal1 may have pleiotropic effects independent of the circadian clock that could impinge on metabolism. Several lines of evidence argue against this hypothesis. First, genes relevant to these metabolic phenotypes display circadian oscillations in their steady-state mRNA levels ( Young et al. 2001 ;  Oishi et al. 2003 ). In addition, the mRNA levels of many of these key proteins are phase-aligned with  Per1  (e.g.,  Enolase 3, Pgam, Transketolase, Lipase, Lpl, Dgat1, Ppar alpha ) or  Per2  (e.g.,  Mod1, Lpl, Pepck, lipin 1 ) (unpublished data). In addition, many of their mRNAs are at lower levels in  Bmal1
 −/−  (e.g.,  Mod1, Pepck, Enolase3, Pgam ) (unpublished data), consistent with a direct role of Bmal1 in their transcription. Thirdly, these metabolic parameters are disrupted in both circadian mutants with the same rank order of potency as the locomotor activity phenotypes  (Bmal1 −/−  > Clock mut ) . Thus, the most parsimonious interpretation is that the observed metabolic deficiencies in the  Bmal1
 −/−  and  Clock mut  mice are due to their roles in the circadian clock, rather than to “off-clock” effects.Per1Enolase 3, Pgam, Transketolase, Lipase, Lpl, Dgat1, Ppar alphaPer2Mod1, Lpl, Pepck, lipin 1Bmal1Mod1, Pepck, Enolase3, Pgam(Bmal1 −/−  > Clock mut )Bmal1Clock mutWe observed that the impact of HF on glucose homeostasis was apparently to emphasize the role of the molecular clock. Diet has previously been shown to interact with peripheral clocks. Changes in feeding shift the circadian pattern of gene expression in the liver, but not in the master clock in the SCN ( Damiola et al. 2000 ), demonstrating the importance of food as a cue to circadian control. Individual constituents of food could also provide discrete stimuli. For example, glucose alone can induce rhythmic gene expression in isolated fibroblasts ( Hirota et al. 2002 ). Thus, dietary composition, the size and timing of a feed might all be expected to interact differentially with an underlying circadian regulation of metabolic control.Alterations in dietary content, the availability of “fast food,” inactivity, and sociocultural factors have all been implicated in the emergence of the metabolic syndrome as a major challenge to the public health ( Zimmet et al. 2001 ). However, while mechanistic integration of the diverse elements of the syndrome has proven elusive, our studies suggest that timing may influence the functional consequences of ingesting a caloric load.Materials and Methods Animals Mice were acclimatized for 2 wk in 12 h light–12 h dark cycles before being subjected to a 36-h period of constant darkness followed by experimentation in darkness. Experimental chronology is measured in CT, subjective day beginning at 7 AM (CT0), and subjective night beginning at 7 PM (CT12). Diet WT and  Clock mut  mice were placed on an HF (Teklad, TD02435) and compared to age-matched WT mice on a regular chow diet (RC). Mice were on RC for 8 wk except for those subjected to FSIGT where they received RC for 11 mo. Body mass composition was measured by dual energy X-ray absorptiometry at 10 mo. Intraperitoneal tolerance Tests were performed as described ( Klaman et al. 2000 ) with a diminution in the glucose bolus (0.1 g/kg). Intravenous glucose tolerance test and minimal modeling The tolerance test was performed as described ( Pacini et al. 2001 ) in unanesthetized mice, and the minimal model of  Bergman et al. (1979)  was applied to the data using MINMOD software ( Boston et al. 2003 ). The derived values were S i , S g , and acute insulin response to glucose, which measures insulin secretion. S i  is the ratio of insulin delivery rate to the interstitium to insulin extraction rate from the interstitium. Long-term feeding of HF to WT mice resulted in imperceptibly small insulin sensitivity values. This could be the consequence of impaired delivery of insulin to the interstitium, exacerbated extraction rate, or a combination of both factors. Insulin secretion is derived from area under the insulin curve, above basal, from 0 to 10 min after glucose infusion; and disposition index, which equals the product of insulin sensitivity multiplied by insulin secretion and measures the degree to which insulin sensitivity can be compensated for by elevated insulin secretion ( Pacini et al. 2001 ). Assay methods Insulin, leptin, corticosterone, and glucagon levels were measured by immunoassays from Crystalchem (Downers Grove, Illinois, United States), ICN Biochemicals (Costa Mesa, California, United States), and Linco Research (St. Charles, Missouri, United States). Plasma glucose was measured by the glucose oxidase method using a glucose analyzer machine for FSIGT and by glucometer for the intraperitoneal tolerance test. PEPCK activity was quantitated by a bioluminescent method ( Wimmer 1988 ). Statistical analysis The significance of differences amongst the tolerance test curves was assessed by distribution-free two-way ANOVA with a Bonferroni correction. FSIGT data were tested by one-way ANOVA with the Kruskal-Wallis test. Paired Student's t-tests were used to perform comparisons of corticosterone levels before and after insulin injection in  Bmal −/−  mice and between WT and  Clock mut  mice. Plasma samples for glucagon analysis were pooled and were thus not compared by a formal statistical analysis. Results are presented as mean ± standard error of the mean (SEM), except for the FSIGT data ( Table 2 ), presented as mean ± fractional standard deviation. Differences were considered significant when  p  < 0.05.
Materials and Methods
Animals Mice were acclimatized for 2 wk in 12 h light–12 h dark cycles before being subjected to a 36-h period of constant darkness followed by experimentation in darkness. Experimental chronology is measured in CT, subjective day beginning at 7 AM (CT0), and subjective night beginning at 7 PM (CT12). Diet WT and  Clock mut  mice were placed on an HF (Teklad, TD02435) and compared to age-matched WT mice on a regular chow diet (RC). Mice were on RC for 8 wk except for those subjected to FSIGT where they received RC for 11 mo. Body mass composition was measured by dual energy X-ray absorptiometry at 10 mo. Intraperitoneal tolerance Tests were performed as described ( Klaman et al. 2000 ) with a diminution in the glucose bolus (0.1 g/kg). Intravenous glucose tolerance test and minimal modeling The tolerance test was performed as described ( Pacini et al. 2001 ) in unanesthetized mice, and the minimal model of  Bergman et al. (1979)  was applied to the data using MINMOD software ( Boston et al. 2003 ). The derived values were S i , S g , and acute insulin response to glucose, which measures insulin secretion. S i  is the ratio of insulin delivery rate to the interstitium to insulin extraction rate from the interstitium. Long-term feeding of HF to WT mice resulted in imperceptibly small insulin sensitivity values. This could be the consequence of impaired delivery of insulin to the interstitium, exacerbated extraction rate, or a combination of both factors. Insulin secretion is derived from area under the insulin curve, above basal, from 0 to 10 min after glucose infusion; and disposition index, which equals the product of insulin sensitivity multiplied by insulin secretion and measures the degree to which insulin sensitivity can be compensated for by elevated insulin secretion ( Pacini et al. 2001 ). Assay methods Insulin, leptin, corticosterone, and glucagon levels were measured by immunoassays from Crystalchem (Downers Grove, Illinois, United States), ICN Biochemicals (Costa Mesa, California, United States), and Linco Research (St. Charles, Missouri, United States). Plasma glucose was measured by the glucose oxidase method using a glucose analyzer machine for FSIGT and by glucometer for the intraperitoneal tolerance test. PEPCK activity was quantitated by a bioluminescent method ( Wimmer 1988 ). Statistical analysis The significance of differences amongst the tolerance test curves was assessed by distribution-free two-way ANOVA with a Bonferroni correction. FSIGT data were tested by one-way ANOVA with the Kruskal-Wallis test. Paired Student's t-tests were used to perform comparisons of corticosterone levels before and after insulin injection in  Bmal −/−  mice and between WT and  Clock mut  mice. Plasma samples for glucagon analysis were pooled and were thus not compared by a formal statistical analysis. Results are presented as mean ± standard error of the mean (SEM), except for the FSIGT data ( Table 2 ), presented as mean ± fractional standard deviation. Differences were considered significant when  p  < 0.05.

Animals Mice were acclimatized for 2 wk in 12 h light–12 h dark cycles before being subjected to a 36-h period of constant darkness followed by experimentation in darkness. Experimental chronology is measured in CT, subjective day beginning at 7 AM (CT0), and subjective night beginning at 7 PM (CT12).
Animals
Mice were acclimatized for 2 wk in 12 h light–12 h dark cycles before being subjected to a 36-h period of constant darkness followed by experimentation in darkness. Experimental chronology is measured in CT, subjective day beginning at 7 AM (CT0), and subjective night beginning at 7 PM (CT12).Diet WT and  Clock mut  mice were placed on an HF (Teklad, TD02435) and compared to age-matched WT mice on a regular chow diet (RC). Mice were on RC for 8 wk except for those subjected to FSIGT where they received RC for 11 mo. Body mass composition was measured by dual energy X-ray absorptiometry at 10 mo.
Diet
WT and  Clock mut  mice were placed on an HF (Teklad, TD02435) and compared to age-matched WT mice on a regular chow diet (RC). Mice were on RC for 8 wk except for those subjected to FSIGT where they received RC for 11 mo. Body mass composition was measured by dual energy X-ray absorptiometry at 10 mo.Clock mutIntraperitoneal tolerance Tests were performed as described ( Klaman et al. 2000 ) with a diminution in the glucose bolus (0.1 g/kg).
Intraperitoneal tolerance
Tests were performed as described ( Klaman et al. 2000 ) with a diminution in the glucose bolus (0.1 g/kg).Intravenous glucose tolerance test and minimal modeling The tolerance test was performed as described ( Pacini et al. 2001 ) in unanesthetized mice, and the minimal model of  Bergman et al. (1979)  was applied to the data using MINMOD software ( Boston et al. 2003 ). The derived values were S i , S g , and acute insulin response to glucose, which measures insulin secretion. S i  is the ratio of insulin delivery rate to the interstitium to insulin extraction rate from the interstitium. Long-term feeding of HF to WT mice resulted in imperceptibly small insulin sensitivity values. This could be the consequence of impaired delivery of insulin to the interstitium, exacerbated extraction rate, or a combination of both factors. Insulin secretion is derived from area under the insulin curve, above basal, from 0 to 10 min after glucose infusion; and disposition index, which equals the product of insulin sensitivity multiplied by insulin secretion and measures the degree to which insulin sensitivity can be compensated for by elevated insulin secretion ( Pacini et al. 2001 ).
Intravenous glucose tolerance test and minimal modeling
The tolerance test was performed as described ( Pacini et al. 2001 ) in unanesthetized mice, and the minimal model of  Bergman et al. (1979)  was applied to the data using MINMOD software ( Boston et al. 2003 ). The derived values were S i , S g , and acute insulin response to glucose, which measures insulin secretion. S i  is the ratio of insulin delivery rate to the interstitium to insulin extraction rate from the interstitium. Long-term feeding of HF to WT mice resulted in imperceptibly small insulin sensitivity values. This could be the consequence of impaired delivery of insulin to the interstitium, exacerbated extraction rate, or a combination of both factors. Insulin secretion is derived from area under the insulin curve, above basal, from 0 to 10 min after glucose infusion; and disposition index, which equals the product of insulin sensitivity multiplied by insulin secretion and measures the degree to which insulin sensitivity can be compensated for by elevated insulin secretion ( Pacini et al. 2001 ).Assay methods Insulin, leptin, corticosterone, and glucagon levels were measured by immunoassays from Crystalchem (Downers Grove, Illinois, United States), ICN Biochemicals (Costa Mesa, California, United States), and Linco Research (St. Charles, Missouri, United States). Plasma glucose was measured by the glucose oxidase method using a glucose analyzer machine for FSIGT and by glucometer for the intraperitoneal tolerance test. PEPCK activity was quantitated by a bioluminescent method ( Wimmer 1988 ).
Assay methods
Insulin, leptin, corticosterone, and glucagon levels were measured by immunoassays from Crystalchem (Downers Grove, Illinois, United States), ICN Biochemicals (Costa Mesa, California, United States), and Linco Research (St. Charles, Missouri, United States). Plasma glucose was measured by the glucose oxidase method using a glucose analyzer machine for FSIGT and by glucometer for the intraperitoneal tolerance test. PEPCK activity was quantitated by a bioluminescent method ( Wimmer 1988 ).Statistical analysis The significance of differences amongst the tolerance test curves was assessed by distribution-free two-way ANOVA with a Bonferroni correction. FSIGT data were tested by one-way ANOVA with the Kruskal-Wallis test. Paired Student's t-tests were used to perform comparisons of corticosterone levels before and after insulin injection in  Bmal −/−  mice and between WT and  Clock mut  mice. Plasma samples for glucagon analysis were pooled and were thus not compared by a formal statistical analysis. Results are presented as mean ± standard error of the mean (SEM), except for the FSIGT data ( Table 2 ), presented as mean ± fractional standard deviation. Differences were considered significant when  p  < 0.05.
Statistical analysis
The significance of differences amongst the tolerance test curves was assessed by distribution-free two-way ANOVA with a Bonferroni correction. FSIGT data were tested by one-way ANOVA with the Kruskal-Wallis test. Paired Student's t-tests were used to perform comparisons of corticosterone levels before and after insulin injection in  Bmal −/−  mice and between WT and  Clock mut  mice. Plasma samples for glucagon analysis were pooled and were thus not compared by a formal statistical analysis. Results are presented as mean ± standard error of the mean (SEM), except for the FSIGT data ( Table 2 ), presented as mean ± fractional standard deviation. Differences were considered significant when  p  < 0.05.Bmal −/−Clock mutp


Making choices is a fundamental aspect of human life. For over a century experimental economists have characterized the decisions people make based on the concept of a utility function. This function increases with increasing desirability of the outcome, and people are assumed to make decisions so as to maximize utility. When utility depends on several variables, indifference curves arise that represent outcomes with identical utility that are therefore equally desirable. Whereas in economics utility is studied in terms of goods and services, the sensorimotor system may also have utility functions defining the desirability of various outcomes. Here, we investigate the indifference curves when subjects experience forces of varying magnitude and duration. Using a two-alternative forced-choice paradigm, in which subjects chose between different magnitude–duration profiles, we inferred the indifference curves and the utility function. Such a utility function defines, for example, whether subjects prefer to lift a 4-kg weight for 30 s or a 1-kg weight for a minute. The measured utility function depends nonlinearly on the force magnitude and duration and was remarkably conserved across subjects. This suggests that the utility function, a central concept in economics, may be applicable to the study of sensorimotor control.

Introduction In real world situations we often have to choose between possible actions that lead to different outcomes. To provide a computational framework for such a decision process, the notion of a utility function is often used ( Neumann and Morgenstern 1944 ). A utility function assigns to each possible action a number that specifies how desirable each outcome is. In the theory of rational choice, it is assumed that subjects will choose the action that leads to the most desirable outcome and thus the highest utility. The economics literature extensively discusses the problem of having a utility function that depends on two or more variables ( Edgeworth 1881 ;  Pareto 1909 ). For example, people may associate a utility with the number of apples and oranges they are offered. There will be combinations of apples and oranges which have equal utility. Having three apples and three oranges could be judged as being equally good as having ten apples and one orange. These two possibilities would form two points along an “indifference curve” in apple–orange space, representing outcomes with identical utility that are, therefore, equally desirable. Such indifference curves have been extensively studied by economists in terms of goods and services (c.f.  Humphrey 1996 ). The sensorimotor system also has to choose between different actions. The utility of actions will depend on two components—the cost associated with performing an action and the desirability of the outcome. Here we characterize the utility function used by the sensorimotor system by measuring the indifference curves for human subjects experiencing short pulses of force. In sensorimotor control, utility functions that depend on several variables occur frequently. Consider, for example, unpacking a car after a snowboarding vacation. We could carry all the suitcases at the same time, reducing the time to unpack but maximizing the weight we have to lift concurrently. At the other extreme we could transport each item individually, which would minimize the magnitude of the force required at the expense of a long unpacking duration. The chosen solution is likely to lie somewhere between these two extremes and may reflect an optimal decision based on a utility function that depends on duration and magnitude of the forces. Once a utility function is specified, the decision problem becomes one of solving an optimal control problem, finding the actions that maximize the utility. A number of studies in the field of optimal sensorimotor control have proposed loss functions (the negative of utility) and derived the optimal actions given these proposed loss functions. For example, the minimum jerk model ( Hogan 1984 ;  Flash and Hogan 1985 ) suggests that people minimize the average squared jerk of the hand (third derivative of position) when making reaching movements. Alternative models have suggested that during reaching people try to minimize the variation of endpoint errors that arise from noise on the motor commands ( Harris and Wolpert 1998 ;  Todorov and Jordan 2002 ). However, these and many other similar studies assume a loss function and compare the predicted behavior with observed behavior, rather than measure the loss function directly. In recent works we have used a statistical approach to infer the loss function, instead of assuming it. We defined the statistics of the errors observed by subjects and showed that they were sensitive to quadratic errors for small errors but that for larger errors they were robust to outliers ( Körding and Wolpert 2004 ). Here we use an alternative approach that is analogous to the approaches used in economics to infer a loss function. Different movements may be associated with different costs or utility. For example, a utility function could assign a numerical value to each possible movement, characterizing how costly it is to the organism. Here, we have examined how the utility associated with producing a force depends on two parameters, the duration and the magnitude of a force profile (see  Materials and Methods  for details). The force profiles were smoothed square waves that could be linearly scaled by the duration of the force,  T,  and the maximum value of the force,  F.  On each trial, subjects experienced two force profiles that differed in both  T  and  F.  They then had to choose which of the two force profiles they would experience again. They were told to choose the force that required the least effort. In this two-alternative forced-choice experiment subjects thus indicated their preference for one combination of  F  and  T  over another combination of  F  and  T.  This allowed us to infer indifference curves: Given the choice of two combinations of  F  and  T  that are on the same indifference curve, subjects will have no preference. The associated utility of these force profiles is thus identical. To obtain a full utility function from a set of utility curves we additionally needed to determine the utility of one indifference curve relative to another. This was achieved by finding “doubling points.” A doubling point is a point on one indifference curve that subjects show no preference for when compared to experiencing a point on another indifference curve twice (that is, two smoothed square waves in quick succession—see  Materials and Methods  for details). Thus we could determine the full utility function.
Introduction
In real world situations we often have to choose between possible actions that lead to different outcomes. To provide a computational framework for such a decision process, the notion of a utility function is often used ( Neumann and Morgenstern 1944 ). A utility function assigns to each possible action a number that specifies how desirable each outcome is. In the theory of rational choice, it is assumed that subjects will choose the action that leads to the most desirable outcome and thus the highest utility. The economics literature extensively discusses the problem of having a utility function that depends on two or more variables ( Edgeworth 1881 ;  Pareto 1909 ). For example, people may associate a utility with the number of apples and oranges they are offered. There will be combinations of apples and oranges which have equal utility. Having three apples and three oranges could be judged as being equally good as having ten apples and one orange. These two possibilities would form two points along an “indifference curve” in apple–orange space, representing outcomes with identical utility that are, therefore, equally desirable. Such indifference curves have been extensively studied by economists in terms of goods and services (c.f.  Humphrey 1996 ). The sensorimotor system also has to choose between different actions. The utility of actions will depend on two components—the cost associated with performing an action and the desirability of the outcome. Here we characterize the utility function used by the sensorimotor system by measuring the indifference curves for human subjects experiencing short pulses of force.In sensorimotor control, utility functions that depend on several variables occur frequently. Consider, for example, unpacking a car after a snowboarding vacation. We could carry all the suitcases at the same time, reducing the time to unpack but maximizing the weight we have to lift concurrently. At the other extreme we could transport each item individually, which would minimize the magnitude of the force required at the expense of a long unpacking duration. The chosen solution is likely to lie somewhere between these two extremes and may reflect an optimal decision based on a utility function that depends on duration and magnitude of the forces. Once a utility function is specified, the decision problem becomes one of solving an optimal control problem, finding the actions that maximize the utility.A number of studies in the field of optimal sensorimotor control have proposed loss functions (the negative of utility) and derived the optimal actions given these proposed loss functions. For example, the minimum jerk model ( Hogan 1984 ;  Flash and Hogan 1985 ) suggests that people minimize the average squared jerk of the hand (third derivative of position) when making reaching movements. Alternative models have suggested that during reaching people try to minimize the variation of endpoint errors that arise from noise on the motor commands ( Harris and Wolpert 1998 ;  Todorov and Jordan 2002 ). However, these and many other similar studies assume a loss function and compare the predicted behavior with observed behavior, rather than measure the loss function directly. In recent works we have used a statistical approach to infer the loss function, instead of assuming it. We defined the statistics of the errors observed by subjects and showed that they were sensitive to quadratic errors for small errors but that for larger errors they were robust to outliers ( Körding and Wolpert 2004 ). Here we use an alternative approach that is analogous to the approaches used in economics to infer a loss function.Different movements may be associated with different costs or utility. For example, a utility function could assign a numerical value to each possible movement, characterizing how costly it is to the organism. Here, we have examined how the utility associated with producing a force depends on two parameters, the duration and the magnitude of a force profile (see  Materials and Methods  for details). The force profiles were smoothed square waves that could be linearly scaled by the duration of the force,  T,  and the maximum value of the force,  F.  On each trial, subjects experienced two force profiles that differed in both  T  and  F.  They then had to choose which of the two force profiles they would experience again. They were told to choose the force that required the least effort. In this two-alternative forced-choice experiment subjects thus indicated their preference for one combination of  F  and  T  over another combination of  F  and  T.  This allowed us to infer indifference curves: Given the choice of two combinations of  F  and  T  that are on the same indifference curve, subjects will have no preference. The associated utility of these force profiles is thus identical. To obtain a full utility function from a set of utility curves we additionally needed to determine the utility of one indifference curve relative to another. This was achieved by finding “doubling points.” A doubling point is a point on one indifference curve that subjects show no preference for when compared to experiencing a point on another indifference curve twice (that is, two smoothed square waves in quick succession—see  Materials and Methods  for details). Thus we could determine the full utility function.T,F.TF.FTFT.FTResults/Discussion In a two-alternative forced-choice paradigm, subjects chose which of two experienced force profiles they preferred to experience a second time (see  Figure 1 ). This allowed us to find a set of force profiles to which subjects showed equal preference and were therefore indifferent. Although subjects experience these force profiles as being physically different, they show no preference in terms of which they wish to experience again. Figure 1 The Experimental Setup The subject's hand position (pink circle) was visible on the screen. The hand movement was restricted to stay within a small area (blue box). The direction of the force is represented by the blue arrow, and the temporal profile of the force is shown by the blue curve. Various hypothesized utility functions predict different choices and thus different indifference lines. The first model we hypothesized was that subjects would minimize the integrated force they are using  (F × T).  This predicts hyperbolas as indifference lines ( Figure 2 A). Alternatively, people could minimize the integrated squared force  (F 2  × T)  ( Figure 2 B). In this case they would prefer long-duration, weak forces to short-duration, strong forces of equal integrated force. Another possible model would be that people would just try to minimize the maximal force they had to produce, regardless of how long they had to hold it  (F)  ( Figure 2 C). Figure 2 Hypothesized and Measured Indifference Curves and Loss Function from a Single Subject (A–C) The predicted indifference lines are shown that minimize (A) the integrated force  (F × T),  (B) the integrated squared force  (F 2  × T),  and (C) the maximal force  (F) . (D) Experimental data from a single subject. The open circles are the reference forces. The blue full circles connected by the black lines represent indifference points. Error bars denote the 95% confidence intervals. Force profiles are illustrated (blue curves for single forces, pink curve for doubling points). (E) Inferred color plot of the loss function (warmer colors represent greater cost). A single subject's results are shown in  Figure 2 D. The reference forces are shown as open circles, while the indifference points are shown as filled circles. The location of the indifference points had relatively small error bars (black 95% confidence intervals). Therefore, this subject showed a preference for points towards the origin compared to those further from the origin (along the blue lines). Joining up such points in force–time space allows us to obtain indifference curves (black lines). For short duration profiles (less than 150 ms), as the duration increased, the force needed to decrease to maintain constant utility. This makes intuitive sense: as the duration of experienced force increases, more effort is required to stabilize the arm. For longer durations (greater than 500 ms), as the duration increased, the force required to maintain equal utility also increased. This means people prefer to experience a 2-s force profile compared to experiencing a 1-s force profile. We explain this counterintuitive result—that increasing both the duration and force can keep the utility constant—in the following way. The shape of the force profiles for all conditions was kept self-similar. This means that force profiles with a longer duration have a slow onset and offset (each is 20% of the total duration). For long durations subjects can, therefore, progressively compensate for the imposed forces as they ramp up slowly, thereby producing less loss. We furthermore measured how much smaller a force profile needed to be (scaled uniformly in duration and force) so that experiencing it twice had the same utility as experiencing the unscaled profile once. For the four open-circle reference points in  Figure 2 D, the pink circles show the corresponding four points that have half the utility. We can thus infer how the loss function changes as the force profiles are scaled ( Figure 2 E; see  Materials and Methods ). Any order-preserving transformation of the utility function will have no effect on subjects' preferences. That means that arbitrary scalings can be applied to the loss function while the optimal behavior remains unchanged. This property of utility function is well known in economics and has led to the idea of ordinal utility ( Pareto 1909 ), in which the ordering of preferences is the key feature of utility. The utility of the first reference point is thus arbitrarily set to be equal to one. The plotted relative utility is the utility function arising from this assumption. The double-hump experiment defines the derivative of the utility, which is interpolated and integrated to obtain the relative utility. To infer the relative utility function ( Figure 2 B), we had to assume local linearity. The loss function shows nonlinear behavior.
 Figure 3  shows the inferred utility function averaged over all the subjects. We can analyze how loss increases along the line connecting the reference points ( F/T  = 44.6). Fitting a model of the form Loss = ( FT ) α  to the data from the double-hump experiments leads to an α of 1.1 ± 0.15 (mean ± SEM over subjects). This α, when fit to the data from all the subjects for each of the four lines, is approximately constant (1.2, 1.0, 0.9, 0.9). The shape of the loss function is highly conserved over the set of subjects. In particular, the effect that indifference curves increase for both very short and long durations is found over the set of subjects. Figure 3 Iso-Loss Contours and Loss Function for the Set of All Subjects The black curves are the iso-loss curves. Error bars denote the standard error of the mean over the population. The color plot represents the inferred loss function (warmer colors represent greater loss) obtained by interpolating the data from the double-hump forces. By applying the methodology developed by economists, we have shown that fundamental properties of the nervous system, such as loss functions, can be inferred by the choices humans make in a sensorimotor task. In general, these loss functions will depend on a large number of factors that were not measured in our experiment. For example, there are subjective emotional components to human decision making ( Sanfey et al. 2003 ). However, parametric variations would allow such multi-dimensional loss functions to be determined. Interestingly, the inferred loss function we report cannot easily be modeled by any simple function of our experimental variables  F  and  T.  However, it is highly conserved across the subjects, suggesting a common underlying mechanism is at work. Moreover, our results suggest that the opposite approach—first hypothesizing a loss function and then predicting human decision making—is likely to miss interesting aspects of the behavior and underlying processes. We are therefore hopeful that the application of economic methods to the study of the nervous system, referred to as neuroeconomics ( Glimcher 2003 ), will continue to provide new insights into the functioning of the central nervous system.
Results/Discussion
In a two-alternative forced-choice paradigm, subjects chose which of two experienced force profiles they preferred to experience a second time (see  Figure 1 ). This allowed us to find a set of force profiles to which subjects showed equal preference and were therefore indifferent. Although subjects experience these force profiles as being physically different, they show no preference in terms of which they wish to experience again.Various hypothesized utility functions predict different choices and thus different indifference lines. The first model we hypothesized was that subjects would minimize the integrated force they are using  (F × T).  This predicts hyperbolas as indifference lines ( Figure 2 A). Alternatively, people could minimize the integrated squared force  (F 2  × T)  ( Figure 2 B). In this case they would prefer long-duration, weak forces to short-duration, strong forces of equal integrated force. Another possible model would be that people would just try to minimize the maximal force they had to produce, regardless of how long they had to hold it  (F)  ( Figure 2 C).(F × T).(F 2  × T)(F)A single subject's results are shown in  Figure 2 D. The reference forces are shown as open circles, while the indifference points are shown as filled circles. The location of the indifference points had relatively small error bars (black 95% confidence intervals). Therefore, this subject showed a preference for points towards the origin compared to those further from the origin (along the blue lines). Joining up such points in force–time space allows us to obtain indifference curves (black lines). For short duration profiles (less than 150 ms), as the duration increased, the force needed to decrease to maintain constant utility. This makes intuitive sense: as the duration of experienced force increases, more effort is required to stabilize the arm. For longer durations (greater than 500 ms), as the duration increased, the force required to maintain equal utility also increased. This means people prefer to experience a 2-s force profile compared to experiencing a 1-s force profile. We explain this counterintuitive result—that increasing both the duration and force can keep the utility constant—in the following way. The shape of the force profiles for all conditions was kept self-similar. This means that force profiles with a longer duration have a slow onset and offset (each is 20% of the total duration). For long durations subjects can, therefore, progressively compensate for the imposed forces as they ramp up slowly, thereby producing less loss.We furthermore measured how much smaller a force profile needed to be (scaled uniformly in duration and force) so that experiencing it twice had the same utility as experiencing the unscaled profile once. For the four open-circle reference points in  Figure 2 D, the pink circles show the corresponding four points that have half the utility. We can thus infer how the loss function changes as the force profiles are scaled ( Figure 2 E; see  Materials and Methods ). Any order-preserving transformation of the utility function will have no effect on subjects' preferences. That means that arbitrary scalings can be applied to the loss function while the optimal behavior remains unchanged. This property of utility function is well known in economics and has led to the idea of ordinal utility ( Pareto 1909 ), in which the ordering of preferences is the key feature of utility. The utility of the first reference point is thus arbitrarily set to be equal to one. The plotted relative utility is the utility function arising from this assumption. The double-hump experiment defines the derivative of the utility, which is interpolated and integrated to obtain the relative utility. To infer the relative utility function ( Figure 2 B), we had to assume local linearity. The loss function shows nonlinear behavior.Figure 3  shows the inferred utility function averaged over all the subjects. We can analyze how loss increases along the line connecting the reference points ( F/T  = 44.6). Fitting a model of the form Loss = ( FT ) α  to the data from the double-hump experiments leads to an α of 1.1 ± 0.15 (mean ± SEM over subjects). This α, when fit to the data from all the subjects for each of the four lines, is approximately constant (1.2, 1.0, 0.9, 0.9). The shape of the loss function is highly conserved over the set of subjects. In particular, the effect that indifference curves increase for both very short and long durations is found over the set of subjects.F/TFTBy applying the methodology developed by economists, we have shown that fundamental properties of the nervous system, such as loss functions, can be inferred by the choices humans make in a sensorimotor task. In general, these loss functions will depend on a large number of factors that were not measured in our experiment. For example, there are subjective emotional components to human decision making ( Sanfey et al. 2003 ). However, parametric variations would allow such multi-dimensional loss functions to be determined. Interestingly, the inferred loss function we report cannot easily be modeled by any simple function of our experimental variables  F  and  T.  However, it is highly conserved across the subjects, suggesting a common underlying mechanism is at work. Moreover, our results suggest that the opposite approach—first hypothesizing a loss function and then predicting human decision making—is likely to miss interesting aspects of the behavior and underlying processes. We are therefore hopeful that the application of economic methods to the study of the nervous system, referred to as neuroeconomics ( Glimcher 2003 ), will continue to provide new insights into the functioning of the central nervous system.FT.Materials and Methods Subjects and the manipulandum After providing written informed consent, five right-handed subjects (aged 20–40 y) participated in this study. The experiments were carried out in accordance with institutional guidelines. A local ethics committee approved the experimental protocols. While seated, subjects held the handle of robotic manipulandum with two degrees of planar freedom. This was a custom-built device (vBot) consisting of a parallelogram constructed mainly from carbon fiber tubes that were driven by rare earth motors via low-friction timing belts. High-resolution incremental encoders were attached to the drive motors to permit accurate computation of the robot's position. Care was taken with the design to ensure it was capable of exerting large end-point forces while still exhibiting high stiffness, low friction, and also low inertia. The robot's motors were run from a pair of switching torque control amplifiers that were interfaced, along with the encoders, to a multifunctional I/O card on a PC using some simple logic to implement safety features. Software control of the robot was achieved by means of a control loop running at 1,000 Hz, in which position and force were measured and desired output force was set. A virtual reality system was used that prevented subjects seeing their hand, and allowed us to present visual images into the plane of the movement (for full details of the setup see  Goodbody and Wolpert 1998 ) (see  Figure 1 A). The force between the subject's hand and the manipulandum was continuously measured using a six-axis force transducer (Nano25; ATI Industrial Automation, Apex, North Carolina, United States) sampled at 1,000 Hz by the control loop. The experiment consisted of trials in which the robot generated force profiles on the subjects' hands. The force profiles experienced were parameterized by their duration  T  in ms and their maximal strength  F  in Newtons. The force profile  f  ( t ) approximated a square profile, but with smooth onset and offset:

 On each trial the subjects experienced two different force profiles and then could choose which of the two profiles to experience for a second time. Using such a forced-choice procedure allowed us to determine the indifference curves. Inferring indifference pairs Subjects saw a starting sphere and two selection spheres (see  Figure 1 A). Each trial started when the subject moved the cursor, representing their hand, into the starting sphere. The trial then had three phases. (1) One of the selection spheres turned green, and subjects were required to place the cursor into this sphere, where they experienced a force profile  F 1 . The subjects then returned the cursor to the starting sphere. (2) The other selection sphere turned green, and subjects were required to place the cursor in that sphere, where they experienced a force profile  F 2 . Subjects then returned the cursor to the starting sphere. (3) Both selection spheres turned green, and subjects were required to choose which of the two spheres to move to, where they would experience the same force associated with that sphere, either  F 1  or  F 2 . Therefore, subjects could decide which force profile,  F 1  or  F 2 ,  to experience a second time. To obtain four indifference curves, we chose four reference profiles that had durations  T  of 200, 300, 400 and 500 ms. The maximal force  F  was chosen for each reference so that the ratio  T/F  had the value 44.6. This gave a maximal force that ranged from 4.5 N for the shortest duration reference to 11.2 N for the longest duration reference. These reference points lie along a straight line in time–force space (see  Figure 2 A, open circles). On each trial, one of the two force profiles,  F 1  or  F 2 ,  was set to be one of the reference forces and the other was a test force. The sphere associated with the reference force was randomized each trial between the left and right locations. To obtain indifference lines, we wished to find points along the radial lines shown in  Figure 2 A to which subjects were indifferent to the four reference points. To obtain these we used a two-alternative forced-choice paradigm in which the test force produced was chosen from one of these lines, which correspond to  T/F  ratios of 2.0, 7.4, 20.0, 44.6 (double-hump, see below), 85.4, 142.1, and 203.0, with the aim of finding the point along the line at which subjects would choose between the reference and test force indifferently (that is, at probability level 0.5). We used an adaptive fitting protocol (QUEST;  Watson and Pelli 1983 ) to find the  p  = 0.5 threshold of a logistic function. The reference points and  T/F  ratio lines were interleaved in a pseudorandom order. Forty trials were performed to obtain each indifference pair. Each reference point, together with the six  T/F  ratio line points that subjects preferred equally, defines an indifference curve. Inferring the loss function The above procedure allowed us to obtain indifference lines—where the utility has equal value. However, to obtain a full utility function we need to join up these lines and determine the relative utility of one indifference line to another. To achieve this we performed a two-alternative forced-choice paradigm in which the reference force was as before, but the test force was selected from the  T/F  = 44.6 line, with the force profile presented twice in succession (the “double hump” force). This condition was run interleaved with the other conditions. We assumed that the utility of experiencing the double hump was twice the utility of a single hump (a linearity assumption). This assumption allowed us to link the reference point to a point of half its utility, further allowing us to linearly interpolate log(utility) between these points to obtain estimates of the loss function between the lines.
Materials and Methods
Subjects and the manipulandum After providing written informed consent, five right-handed subjects (aged 20–40 y) participated in this study. The experiments were carried out in accordance with institutional guidelines. A local ethics committee approved the experimental protocols. While seated, subjects held the handle of robotic manipulandum with two degrees of planar freedom. This was a custom-built device (vBot) consisting of a parallelogram constructed mainly from carbon fiber tubes that were driven by rare earth motors via low-friction timing belts. High-resolution incremental encoders were attached to the drive motors to permit accurate computation of the robot's position. Care was taken with the design to ensure it was capable of exerting large end-point forces while still exhibiting high stiffness, low friction, and also low inertia. The robot's motors were run from a pair of switching torque control amplifiers that were interfaced, along with the encoders, to a multifunctional I/O card on a PC using some simple logic to implement safety features. Software control of the robot was achieved by means of a control loop running at 1,000 Hz, in which position and force were measured and desired output force was set. A virtual reality system was used that prevented subjects seeing their hand, and allowed us to present visual images into the plane of the movement (for full details of the setup see  Goodbody and Wolpert 1998 ) (see  Figure 1 A). The force between the subject's hand and the manipulandum was continuously measured using a six-axis force transducer (Nano25; ATI Industrial Automation, Apex, North Carolina, United States) sampled at 1,000 Hz by the control loop. The experiment consisted of trials in which the robot generated force profiles on the subjects' hands. The force profiles experienced were parameterized by their duration  T  in ms and their maximal strength  F  in Newtons. The force profile  f  ( t ) approximated a square profile, but with smooth onset and offset:

 On each trial the subjects experienced two different force profiles and then could choose which of the two profiles to experience for a second time. Using such a forced-choice procedure allowed us to determine the indifference curves. Inferring indifference pairs Subjects saw a starting sphere and two selection spheres (see  Figure 1 A). Each trial started when the subject moved the cursor, representing their hand, into the starting sphere. The trial then had three phases. (1) One of the selection spheres turned green, and subjects were required to place the cursor into this sphere, where they experienced a force profile  F 1 . The subjects then returned the cursor to the starting sphere. (2) The other selection sphere turned green, and subjects were required to place the cursor in that sphere, where they experienced a force profile  F 2 . Subjects then returned the cursor to the starting sphere. (3) Both selection spheres turned green, and subjects were required to choose which of the two spheres to move to, where they would experience the same force associated with that sphere, either  F 1  or  F 2 . Therefore, subjects could decide which force profile,  F 1  or  F 2 ,  to experience a second time. To obtain four indifference curves, we chose four reference profiles that had durations  T  of 200, 300, 400 and 500 ms. The maximal force  F  was chosen for each reference so that the ratio  T/F  had the value 44.6. This gave a maximal force that ranged from 4.5 N for the shortest duration reference to 11.2 N for the longest duration reference. These reference points lie along a straight line in time–force space (see  Figure 2 A, open circles). On each trial, one of the two force profiles,  F 1  or  F 2 ,  was set to be one of the reference forces and the other was a test force. The sphere associated with the reference force was randomized each trial between the left and right locations. To obtain indifference lines, we wished to find points along the radial lines shown in  Figure 2 A to which subjects were indifferent to the four reference points. To obtain these we used a two-alternative forced-choice paradigm in which the test force produced was chosen from one of these lines, which correspond to  T/F  ratios of 2.0, 7.4, 20.0, 44.6 (double-hump, see below), 85.4, 142.1, and 203.0, with the aim of finding the point along the line at which subjects would choose between the reference and test force indifferently (that is, at probability level 0.5). We used an adaptive fitting protocol (QUEST;  Watson and Pelli 1983 ) to find the  p  = 0.5 threshold of a logistic function. The reference points and  T/F  ratio lines were interleaved in a pseudorandom order. Forty trials were performed to obtain each indifference pair. Each reference point, together with the six  T/F  ratio line points that subjects preferred equally, defines an indifference curve. Inferring the loss function The above procedure allowed us to obtain indifference lines—where the utility has equal value. However, to obtain a full utility function we need to join up these lines and determine the relative utility of one indifference line to another. To achieve this we performed a two-alternative forced-choice paradigm in which the reference force was as before, but the test force was selected from the  T/F  = 44.6 line, with the force profile presented twice in succession (the “double hump” force). This condition was run interleaved with the other conditions. We assumed that the utility of experiencing the double hump was twice the utility of a single hump (a linearity assumption). This assumption allowed us to link the reference point to a point of half its utility, further allowing us to linearly interpolate log(utility) between these points to obtain estimates of the loss function between the lines.

Subjects and the manipulandum After providing written informed consent, five right-handed subjects (aged 20–40 y) participated in this study. The experiments were carried out in accordance with institutional guidelines. A local ethics committee approved the experimental protocols. While seated, subjects held the handle of robotic manipulandum with two degrees of planar freedom. This was a custom-built device (vBot) consisting of a parallelogram constructed mainly from carbon fiber tubes that were driven by rare earth motors via low-friction timing belts. High-resolution incremental encoders were attached to the drive motors to permit accurate computation of the robot's position. Care was taken with the design to ensure it was capable of exerting large end-point forces while still exhibiting high stiffness, low friction, and also low inertia. The robot's motors were run from a pair of switching torque control amplifiers that were interfaced, along with the encoders, to a multifunctional I/O card on a PC using some simple logic to implement safety features. Software control of the robot was achieved by means of a control loop running at 1,000 Hz, in which position and force were measured and desired output force was set. A virtual reality system was used that prevented subjects seeing their hand, and allowed us to present visual images into the plane of the movement (for full details of the setup see  Goodbody and Wolpert 1998 ) (see  Figure 1 A). The force between the subject's hand and the manipulandum was continuously measured using a six-axis force transducer (Nano25; ATI Industrial Automation, Apex, North Carolina, United States) sampled at 1,000 Hz by the control loop. The experiment consisted of trials in which the robot generated force profiles on the subjects' hands. The force profiles experienced were parameterized by their duration  T  in ms and their maximal strength  F  in Newtons. The force profile  f  ( t ) approximated a square profile, but with smooth onset and offset:

 On each trial the subjects experienced two different force profiles and then could choose which of the two profiles to experience for a second time. Using such a forced-choice procedure allowed us to determine the indifference curves.
Subjects and the manipulandum
After providing written informed consent, five right-handed subjects (aged 20–40 y) participated in this study. The experiments were carried out in accordance with institutional guidelines. A local ethics committee approved the experimental protocols.While seated, subjects held the handle of robotic manipulandum with two degrees of planar freedom. This was a custom-built device (vBot) consisting of a parallelogram constructed mainly from carbon fiber tubes that were driven by rare earth motors via low-friction timing belts. High-resolution incremental encoders were attached to the drive motors to permit accurate computation of the robot's position. Care was taken with the design to ensure it was capable of exerting large end-point forces while still exhibiting high stiffness, low friction, and also low inertia.The robot's motors were run from a pair of switching torque control amplifiers that were interfaced, along with the encoders, to a multifunctional I/O card on a PC using some simple logic to implement safety features. Software control of the robot was achieved by means of a control loop running at 1,000 Hz, in which position and force were measured and desired output force was set.A virtual reality system was used that prevented subjects seeing their hand, and allowed us to present visual images into the plane of the movement (for full details of the setup see  Goodbody and Wolpert 1998 ) (see  Figure 1 A). The force between the subject's hand and the manipulandum was continuously measured using a six-axis force transducer (Nano25; ATI Industrial Automation, Apex, North Carolina, United States) sampled at 1,000 Hz by the control loop.The experiment consisted of trials in which the robot generated force profiles on the subjects' hands. The force profiles experienced were parameterized by their duration  T  in ms and their maximal strength  F  in Newtons. The force profile  f  ( t ) approximated a square profile, but with smooth onset and offset:TFftOn each trial the subjects experienced two different force profiles and then could choose which of the two profiles to experience for a second time. Using such a forced-choice procedure allowed us to determine the indifference curves.Inferring indifference pairs Subjects saw a starting sphere and two selection spheres (see  Figure 1 A). Each trial started when the subject moved the cursor, representing their hand, into the starting sphere. The trial then had three phases. (1) One of the selection spheres turned green, and subjects were required to place the cursor into this sphere, where they experienced a force profile  F 1 . The subjects then returned the cursor to the starting sphere. (2) The other selection sphere turned green, and subjects were required to place the cursor in that sphere, where they experienced a force profile  F 2 . Subjects then returned the cursor to the starting sphere. (3) Both selection spheres turned green, and subjects were required to choose which of the two spheres to move to, where they would experience the same force associated with that sphere, either  F 1  or  F 2 . Therefore, subjects could decide which force profile,  F 1  or  F 2 ,  to experience a second time. To obtain four indifference curves, we chose four reference profiles that had durations  T  of 200, 300, 400 and 500 ms. The maximal force  F  was chosen for each reference so that the ratio  T/F  had the value 44.6. This gave a maximal force that ranged from 4.5 N for the shortest duration reference to 11.2 N for the longest duration reference. These reference points lie along a straight line in time–force space (see  Figure 2 A, open circles). On each trial, one of the two force profiles,  F 1  or  F 2 ,  was set to be one of the reference forces and the other was a test force. The sphere associated with the reference force was randomized each trial between the left and right locations. To obtain indifference lines, we wished to find points along the radial lines shown in  Figure 2 A to which subjects were indifferent to the four reference points. To obtain these we used a two-alternative forced-choice paradigm in which the test force produced was chosen from one of these lines, which correspond to  T/F  ratios of 2.0, 7.4, 20.0, 44.6 (double-hump, see below), 85.4, 142.1, and 203.0, with the aim of finding the point along the line at which subjects would choose between the reference and test force indifferently (that is, at probability level 0.5). We used an adaptive fitting protocol (QUEST;  Watson and Pelli 1983 ) to find the  p  = 0.5 threshold of a logistic function. The reference points and  T/F  ratio lines were interleaved in a pseudorandom order. Forty trials were performed to obtain each indifference pair. Each reference point, together with the six  T/F  ratio line points that subjects preferred equally, defines an indifference curve.
Inferring indifference pairs
Subjects saw a starting sphere and two selection spheres (see  Figure 1 A). Each trial started when the subject moved the cursor, representing their hand, into the starting sphere. The trial then had three phases. (1) One of the selection spheres turned green, and subjects were required to place the cursor into this sphere, where they experienced a force profile  F 1 . The subjects then returned the cursor to the starting sphere. (2) The other selection sphere turned green, and subjects were required to place the cursor in that sphere, where they experienced a force profile  F 2 . Subjects then returned the cursor to the starting sphere. (3) Both selection spheres turned green, and subjects were required to choose which of the two spheres to move to, where they would experience the same force associated with that sphere, either  F 1  or  F 2 . Therefore, subjects could decide which force profile,  F 1  or  F 2 ,  to experience a second time.F 1F 2F 1F 2F 1F 2 ,To obtain four indifference curves, we chose four reference profiles that had durations  T  of 200, 300, 400 and 500 ms. The maximal force  F  was chosen for each reference so that the ratio  T/F  had the value 44.6. This gave a maximal force that ranged from 4.5 N for the shortest duration reference to 11.2 N for the longest duration reference. These reference points lie along a straight line in time–force space (see  Figure 2 A, open circles).TFT/FOn each trial, one of the two force profiles,  F 1  or  F 2 ,  was set to be one of the reference forces and the other was a test force. The sphere associated with the reference force was randomized each trial between the left and right locations. To obtain indifference lines, we wished to find points along the radial lines shown in  Figure 2 A to which subjects were indifferent to the four reference points. To obtain these we used a two-alternative forced-choice paradigm in which the test force produced was chosen from one of these lines, which correspond to  T/F  ratios of 2.0, 7.4, 20.0, 44.6 (double-hump, see below), 85.4, 142.1, and 203.0, with the aim of finding the point along the line at which subjects would choose between the reference and test force indifferently (that is, at probability level 0.5). We used an adaptive fitting protocol (QUEST;  Watson and Pelli 1983 ) to find the  p  = 0.5 threshold of a logistic function. The reference points and  T/F  ratio lines were interleaved in a pseudorandom order. Forty trials were performed to obtain each indifference pair. Each reference point, together with the six  T/F  ratio line points that subjects preferred equally, defines an indifference curve.F 1F 2 ,T/FpT/FT/FInferring the loss function The above procedure allowed us to obtain indifference lines—where the utility has equal value. However, to obtain a full utility function we need to join up these lines and determine the relative utility of one indifference line to another. To achieve this we performed a two-alternative forced-choice paradigm in which the reference force was as before, but the test force was selected from the  T/F  = 44.6 line, with the force profile presented twice in succession (the “double hump” force). This condition was run interleaved with the other conditions. We assumed that the utility of experiencing the double hump was twice the utility of a single hump (a linearity assumption). This assumption allowed us to link the reference point to a point of half its utility, further allowing us to linearly interpolate log(utility) between these points to obtain estimates of the loss function between the lines.
Inferring the loss function
The above procedure allowed us to obtain indifference lines—where the utility has equal value. However, to obtain a full utility function we need to join up these lines and determine the relative utility of one indifference line to another. To achieve this we performed a two-alternative forced-choice paradigm in which the reference force was as before, but the test force was selected from the  T/F  = 44.6 line, with the force profile presented twice in succession (the “double hump” force). This condition was run interleaved with the other conditions. We assumed that the utility of experiencing the double hump was twice the utility of a single hump (a linearity assumption). This assumption allowed us to link the reference point to a point of half its utility, further allowing us to linearly interpolate log(utility) between these points to obtain estimates of the loss function between the lines.T/F