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This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators. [ABSTRACT FROM AUTHOR]

In this paper we consider the question when a triangularizable semigroup S of positive compact ideal-triangularizable operators on an order continuous Banach lattice X is ideal-triangularizable. We prove that triangularizability always implies ideal-triangularizability iff X contains at most one atom. Under this condition we connect ideal-triangularizability of S with spectral properties of S. Surprisingly, S is idealtriangularizable iff the spectral radius is subadditive on S iff the spectral radius is submultiplicative on S. We also consider a pair of positive compact operators A and T with the property that A has a T-stable spectrum.

During late adolescence, the brain undergoes ontogenic organization altering subcortical-cortical circuitry. This includes regions implicated in pain chronicity, and thus alterations in the adolescent ontogenic organization could predispose to pain chronicity in adulthood - however, evidence is lacking. Using resting-state functional magnetic resonance imaging from a large European longitudinal adolescent cohort and an adult cohort with and without chronic pain, we examined links between painful symptoms and brain connectivity. During late adolescence, thalamo-, caudate-, and red nucleus-cortical connectivity were positively and subthalamo-cortical connectivity negatively associated with painful symptoms. Thalamo-cortical connectivity, but also subthalamo-cortical connectivity, was increased in adults with chronic pain compared to healthy controls. Our results indicate a shared basis in basothalamo-cortical circuitries between adolescent painful symptomatology and adult pain chronicity, with the subthalamic pathway being differentially involved, potentially due to a hyperconnected thalamo-cortical pathway in chronic pain and ontogeny-driven organization. This can inform neuromodulation-based prevention and early intervention., Competing Interests: Dr Banaschewski served in an advisory or consultancy role for Lundbeck, Medice, Neurim Pharmaceuticals, Oberberg GmbH, Shire. He received conference support or speaker’s fee by Lilly, Medice, Novartis and Shire. He has been involved in clinical trials conducted by Shire & Viforpharma. He received royalties from Hogrefe, Kohlhammer, CIP Medien, Oxford University Press. The present work is unrelated to the above grants and relationships. Dr Barker has received honoraria from General Electric Healthcare for teaching on scanner programming courses. Dr Poustka served in an advisory or consultancy role for Roche and Viforpharm and received speaker’s fee by Shire. She received royalties from Hogrefe, Kohlhammer and Schattauer. The present work is unrelated to the above grants and relationships. The other authors report no biomedical financial interests or potential conflicts of interest., (© 2024 The Authors.)

Background: Chronic pain is a common health problem, for which the treatment is complex and challenging. Non-invasive brain stimulation techniques, specifically transcranial alternating current stimulation (tACS), show promise as a well-tolerated new therapeutic modality with few side effects. This is supported by growing evidence of an association between altered neuronal oscillations and chronic pain. However, to date, only a handful of studies with variable methodology have evaluated tACS for potential applicability to patients with chronic pain., Objectives: Presentation and discussion of the evidence thus far, evaluation of a potential therapeutic benefit for chronic pain patients., Materials and Methods: Literature search in MEDLINE, Embase, Cochrane Library, and Google Scholar databases., Results: To date, tACS for chronic pain therapy has been investigated in only three studies with very different methodological approaches and quality., Discussion: These data currently do not provide sufficient evidence for the therapeutic use of tACS for chronic pain therapy. Future studies may address the question of a therapeutic benefit of tACS for this indication utilizing improved stimulation techniques and considering existing recommendations for the design and conduct of tACS studies., (© 2022. The Author(s), under exclusive licence to Springer Medizin Verlag GmbH, ein Teil von Springer Nature.)

Social interactions affect individual behaviours, preferences, and attitudes. This is also critical in the context of experiencing pain and expressing pain behaviours, and may relate to learned emotional responses. In this respect, individual variability in the medial prefrontal cortex (mPFC), which is involved in adjusting an organism's behaviour to its environment by evaluating and interpreting information within the context of past experiences, is important. It is critical for selecting suitable behavioural responses within a social environment and may reinforce maladaptation in chronic pain. In our study, we used brain imaging during appetitive and aversive pavlovian conditioning in persons with chronic back pain (CBP), subacute back pain (SABP), and healthy controls (HC), together with information on spouse responses to pain behaviours. We also examined the relationship of these responses with pain-related interference in the patients. Our findings yielded a significant negative association between mPFC responses to appetitive and aversive learning in CBP. We also observed a significant negative association for mPFC responses during aversive learning and distracting spouse responses, and a significant positive association between mPFC responses during appetitive learning and solicitous spouse responses in CBP. Both significantly predicted pain-related interference in the CBP group (explained variance up to 53%). Significant associations were not found for SABP or HC. Our findings support an association between appetitive and aversive pavlovian learning, related brain circuits and spouse responses to pain in CBP, where appetitive and aversive learning processes seem to be differentially involved. This can inform prevention and early intervention in a mechanistic approach., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Author(s).)

Connectivity between the nucleus accumbens (NAc) and ventromedial prefrontal cortex (vmPFC) and reward learning independently predict the transition from acute to chronic back pain (CBP). However, how these predictors are related remains unclear. Using functional magnetic resonance imaging, we investigate NAc- and vmPFC-dependent reward learning in 50 patients with subacute back pain (SABP) and follow them over 6 months. Additionally, we compare 29 patients with CBP and 29 pain-free controls to characterize mechanisms of reward learning in the chronic stage. We find that the learning-related updating of the value of reinforcement (prediction error) in the NAc predicts the transition to chronicity. In CBP, compared with controls, vmPFC responses to this prediction error signal are decreased, but increased during a discriminative stimulus. Distinct processes of reward learning in the vmPFC and NAc characterize the development and maintenance of CBP. These could be targeted for the prevention and treatment of chronic pain., Competing Interests: Declaration of interests The authors declare no competing interests., (Copyright © 2022 The Authors. Published by Elsevier Inc. All rights reserved.)

It has been well-documented that the brain changes in states of chronic pain. Less is known about changes in the brain that predict the transition from acute to chronic pain. Evidence from neuroimaging studies suggests a shift from brain regions involved in nociceptive processing to corticostriatal brain regions that are instrumental in the processing of reward and emotional learning in the transition to the chronic state. In addition, dysfunction in descending pain modulatory circuits encompassing the periaqueductal gray and the rostral anterior cingulate cortex may also be a key risk factor for pain chronicity. Although longitudinal imaging studies have revealed potential predictors of pain chronicity, their causal role has not yet been determined. Here we review evidence from studies that involve non-invasive brain stimulation to elucidate to what extent they may help to elucidate the brain circuits involved in pain chronicity. Especially, we focus on studies using non-invasive brain stimulation techniques [e.g., transcranial magnetic stimulation (TMS), particularly its repetitive form (rTMS), transcranial alternating current stimulation (tACS), and transcranial direct current stimulation (tDCS)] in the context of musculoskeletal pain chronicity. We focus on the role of the motor cortex because of its known contribution to sensory components of pain via thalamic inhibition, and the role of the dorsolateral prefrontal cortex because of its role on cognitive and affective processing of pain. We will also discuss findings from studies using experimentally induced prolonged pain and studies implicating the DLPFC, which may shed light on the earliest transition phase to chronicity. We propose that combined brain stimulation and imaging studies might further advance mechanistic models of the chronicity process and involved brain circuits. Implications and challenges for translating the research on mechanistic models of the development of chronic pain to clinical practice will also be addressed., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2021 Kandić, Moliadze, Andoh, Flor and Nees.)