regression - What is a strict definition of U-shaped relationship? - Cross Validated
A U-shape is a function with weighted average derivative negative until a point, and Simonsohn's Two-Lines: The First Valid Test of U-Shaped Relationships). . to refer to the same thing that "U-shape" and "inverted U-shape" are used to refer to: In computer science this is sometimes called "bitonic". Despite the great popularity of the inverted-U curve, or the Yerkes-Dodson law, to describe .. Mean comparisons on the scores for each variable confirmed that the .. MemStick), the Swiss National Science Foundation (), and . The so called “inverted U-shaped dose-effect curve” (IUSDEC) is a nonlinear . The inverted-U shape relationship between corticosteroids and memory led to .. Articles from Nonlinearity in Biology, Toxicology, Medicine are provided here.
As was done in previous studies Ma et al. It typically shows right-hemisphere preponderance and usually maximizes over right prefrontal cortex for a recent review, see Brunia et al.
Previous studies demonstrated that a larger SPN indicated an enhanced expectation toward the outcome and intensified intrinsic motivation toward the task e. In the current research, the near miss condition of the game provides optimal challenge compared with the complete defeat condition, which bears more motivational relevance to participants and is predicted to be more intrinsically motivating and engaging.
Thus, we assume that participants will form greater anticipation toward feedback during the near miss match, resulting in an enlarged SPN.
Inverted U - Wikipedia
A potential limitation of our previous study is that we neglected to collect subjective data, which may serve to support our electrophysiological findings Meng et al. To address this limitation, after the formal experiment, participants were asked to report their enjoyment level, effort expended as well as the extent they cared about task performance for each round of the game.
Thus, we predict participants to enjoy more, invest greater effort and care more about task performance in the near miss condition. As to the task performance, previous studies consistently argued that intrinsic motivation plays a significant role in producing adaptive outcomes e.
For instance, by measuring the perception of time in the computer games, it was found that participants had a greater intrinsic motivation and performed better in the adaptive playing mode than in the overload condition Keller and Bless, Thus, we predict participants to be more committed to experimental tasks and to perform better in the near miss condition.
They were all native Chinese speakers, had normal or corrected-to-normal vision, and did not have any history of neurological disorder or mental diseases.
This research was approved by the Neuromanagement Lab Ethics Committee at Zhejiang University, and all participants provided written informed consent before the experiment. A male experimenter acting as the confederate was matched with male participants as pairs to take part in the formal experiment. Data of two participants were discarded because at least one of the experimental conditions were unsuccessfully manipulated.
Thus, there were eighteen valid participants for the final analysis. Stimuli and procedure Before the experiment, the participant was briefly introduced to his co-player the confederate face-to-face.
Then experimental environment and facilities unfamiliar to participants were introduced. We strictly tracked these procedures to make the participants believe in the authenticity of the two-player online game. After that, the two players were led to take seats in separate rooms which are dim, sound-attenuated and electrically shielded.
Experimental stimuli were presented on the computer screen at a distance of cm away from subjects, with a visual angle of 7.
Stimuli, recording triggers, and response data were presented and recorded by E-Prime 2. Should we still consider such a function a U-shape? In my opinion, such a discussion should be had when you define what a U-shape means to you for your application, and when you specify your null hypothesis. Arbitrary decisions are necessary with this proposed framework. The important thing is to be open about them and check how sensitive results are to changes and to challenge others to do the same.
In addition to stating the null hypothesis, as always you should state the assumptions you rely on. For example, a common assumption is that the regression function is either U-shaped on monotone. The Appropriate Test for a U-Shaped Relationship", where they propose an improvement on the vanilla OLS quadratic test by testing that the derivative of a specified functional form is negative at the beginning of the range, and positive at the end.
An additional point to consider is: Do you want a test that rejects the null hypothesis because of a small violation of U-shapedness? A quite clear-cut proof that the memory impairment induced by high doses of epinephrine is due to opioid peptide release was the demonstration that this effect was blocked by naloxone Introini-Collison and McGaugh On the other hand, the systemic administration of opioid antagonists, by itself, caused memory enhancement, either in a passive avoidance test in the mouse Introini-Collison and McGaugh or on recognition memory in the monkey Aigner and Mishkin It has been suggested that epinephrine effects on memory might be mediated, at least in part, by the release of glucose Gold, Post-training systemic injections of glucose produce non-linear, dose-dependent effects on inhibitory avoidance retention, similar to those reported after epinephrine administration Gold, Moreover, plasma levels of glucose measured shortly after training vary according to the footshock intensity used in training.
Interestingly, the systemic, post-training glucose administration enhanced the retention performance of a habituation response in the open-field in the mouse, whereas insulin administration acted in the opposite way. Since glucose readily enters the brain, it may be that glucose affects memory by directly affecting brain glucoreceptors Oomura et al.
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Moreover post-training cerebroventricular glucose administration produces dose-dependent effects on retention Lee et al. Facilitation of memory processes is reported when amphetamine is administered shortly after a training experience.
In a manner similar to several other adrenergic agents, systemic amphetamine administration exerts a IUSDEC action on memory processes. In early studies memory facilitation was reported after post-training amphetamine administration at low dosages: Similarly, high doses of post-training amphetamine resulted in memory disruption in a single-trial inhibitory avoidance paradigm Weissman, Amphetamine acts through peripheral catecholamine mechanisms: It is well known that emotional arousal also activates the hypothalamic-pituitary-adrenocortical axis, elevating plasma levels of corticosterone.
Ample evidence indicates that glucocorticoids influence long-term memory consolidation De Kloet et al. It has been shown that their effects on memory follow an inverted-U shape relationship. Acute corticosterone administration influences the spatial memory deficit induced by adrenalectomy in adult rats in a biphasic way McCormick et al.
Acute post-training administration of low doses of glucocorticoids enhances memory consolidation, in a manner highly similar to that of epinephrine on spatial memory Sandi et al. On the other hand, it appears that adrenergic and glucocorticoid hormonal systems interact, influencing memory consolidation.
In fact, blockade of the corticosterone stress response, by means of the corticosterone synthesis inhibitor metyrapone, prevents the inhibitory avoidance retention enhancement induced by post-training epinephrine injections or exposure to psychological stress Roozendaal et al.
In humans, the IUSDEC relationship reported between glucocorticoid levels and cognitive function was explained as due to increased arousal.
Circadian variations of the effect of corticosterone oral administration on a free recall test in young humans were measured Fehm-Wolfsdorf et al. Corticosterone administration suppressed the increased cognitive performance in the morning when endogenous corticosterone levels are at their peak, while it had no effect on cognitive performance when administered at night, when corticosterone is at the lowest concentration.
Probably the high endogenous corticosterone levels in the morning corresponded to the peak of the inverted-U shape function between corticosterone levels and cognitive performance, and the corticosterone administration at that time shifted the performance towards a decrease. On the contrary, the corticosterone administration in the evening at low endogenous corticosterone levels may not have been sufficient to increase cognitive performance toward the peak of the inverted-U shape function influencing the processes of arousal and selective attention.
The inverted-U shape relationship between corticosteroids and memory led to the question of whether this process involves opposing or synergic processes that could be mediated by the two types of adrenal steroid receptors reported to exist in the brain: When the performance in the Y-maze of rats administered with either Type I or Type II receptor antagonists was measured, only the Type II antagonist-treated group showed impaired spatial memory performance Conrad et al.
Successively the authors showed that if a IUSDEC explains the results obtained with memory performance at different corticosterone doses, it may only be related to Type II receptor activation Conrad et al. The report that glucocorticoid effects on memory consolidation enhancement depend on the emotionally arousing content of the administered stimulation Sandi, ; Buchanan and Lovallois consistent with extensive evidence indicating that noradrenergic activation in the amygdala is involved in mediating glucocorticoid effects on memory consolidation De Quervain et al.
The infusion into the basolateral amygdala, immediately after training, of the specific Type II agonist RU enhances retention performance while the infusion of the Type II antagonist RU impairs retention performance Roozendaal and McGaugh Selective lesions of this nucleus block retention enhancement induced by post-training systemic injections of dexamethasone Roozendaal and McGaugh Thus glucocorticoid effects on memory consolidation depend on basolateral amygdala function.
Moreover, noradrenergic cell groups of the nucleus of the solitary tract and of the locus coeruleus express high densities of Type II receptors Harfstrand et al.
Post-training activation of these receptors on noradrenergic cell groups in the nucleus of the solitary tract induces memory enhancement Roozendaal et al.
Not all agents that influence memory, presumably acting on arousal levels, act through peripheral adrenergic mechanisms. Post-training subcutaneous injections of ACTH affect later avoidance retention performance. The effects on memory are dose dependent; immediate post-training, systemic administrations of moderate doses of ACTH enhance, and higher doses impair memory storage in a passive avoidance paradigm in the rat Gold and van Buskirk It has been shown that ACTH interaction with the level of training-related stress is quite similar to that of amines: But, on the other hand, systemic ACTH injections do not produce reliable changes in epinephrine and norepinephrine plasma levels.
ACTH, then, does not initiate adrenomedullary or sympathetic activity which would normally follow a footshock, and this hormone must therefore act through other probably central mechanisms McCarty and Gold, ACTH cerebroventricular administration either post-training or 1 h before retention testing enhanced or disrupted the passive avoidance response in the rat according to the dosage-arousal levels Sahgal et al.
Similarly, vasopressin effects on learning and memory were discussed as due to emotional arousal level modulation Sahgal ; Ambrogi Lorenzini et al.
Learning under stress: The inverted-U-shape function revisited
Indeed, the initially reported results showing that post-training vasopressin administration facilitated memory processes in a dose-dependent manner, were presented as proof that vasopressin peculiarly enhanced mnemonic capacity De Wied et al.
Later investigations showed that this effect presumably was due to arousal modifications. Thus the amine levels-footshock intensity relationship suggests that if an animal is in a state of low arousal before vasopressin treatment, then an increase in arousal will facilitate performance. However, if the animal is in an optimal or high arousal state, a further increase in arousal will impair performance.
It was proposed that vasopressin may be involved in the selection of a high arousal state, or in the regulation of arousal by the noradrenergic dorsal bundle Sahgal Finally, oxytocin, another neurosecretory product of the hypothalamo-neurohypophyseal system, appears to have effects opposite to those of vasopressin.
Oxytocin impairs passive avoidance performance after post-trial systemic administration and this effect is dose-dependent in a biphasic manner Bohus et al. As stated in the Introduction, in several papers the hypothesis of a relationship between IUSDEC and the emotional arousal state is not discussed or presented. The post-training intracerebroventricular administration of 2-deoxy-D-galactose a compound antagonizing glycoprotein fucosylation disrupts the retention of a passive avoidance response in the rat Ambrogi Lorenzini et al.
The pre-training systemic administration of the nootrope minaprine enhances an active avoidance response in the rat Ambrogi Lorenzini et al. The intracerebroventricular administration of the neuropeptide PACAP enhances the passive avoidance response in the rat Sacchetti et al.
The same compound elicites a similar dose-response effect on the excitability of an in vitro rat hippocampal slice preparation Roberto et al. In some instances that finding was explained by simply suggesting down-regulation or tolerance. In the case of cholinesterase inhibitors, the hypothesis that the activation of presynaptic autoreceptors may play a role in reducing the activity of these compounds was advanced Braida et al.
In an early study it was reported that systemic post-training administration of physostigmine affects memory processing of an appetitive maze learning task in rats, again following a IUSDEC trend Stratton and Petrinovich Similarly, more recent acetylcholinesterase inhibitors MF, MF were found to antagonize scopolamine-induced amnesia of spatial memory tasks in the rat at low but not at high dosages pre-trial oral administration Braida et al.
Thus, in the present paper it is not possible to clarify all the unanswered questions, possibly related to facets and mechanisms not yet adequately investigated.
Learning under stress: The inverted-U-shape function revisited
This non-linear relationship has been reported for many active compounds, in several learning paradigms, in some animal species and does not depend on either administration route systemic or endocerebral or on administration time before or after training. The IUSDEC response is possibly a multifactorial phenomenon, and the single components may not be easy to isolate experimentally. Nevertheless some mechanisms, at least, have been well studied.
On the other hand in many instances the IUSDEC in learning and memory was not attributed to emotional arousal modifications. Although this hypothesis was not discussed in these papers, arousal levels may be involved in even these circumstances, since there is no experimental evidence which completely excludes it.
The fact that the IUSDEC has been reported equally for compounds exhibiting quite opposite positive or negative effects on learning and memory is not of secondary importance. It may be more difficult to propose this hypothesis for compounds which cause amnesia at low dosages, the effect disappearing at higher ones. But it must be remembered, when considering this hypothesis, that there are several mechanisms, simultaneous or otherwise, which concur to elevate or decrease emotional arousal levels adrenal output, pituitary hormones, interoceptive and exteroceptive environmental stimulation and activation of the reticular formation, emotional commutations of the sensorial input.