I suspect this is just a correlation thing rather than causative. For your brain to function properly you need to have a good diet, sleep and exercise. If you aren't doing those then your brain isn't going to work properly showing up as things like memory dysfunction and depression.

We have studies that show the causal effect of exercise and have a ideas on the mechanisms why

>Aerobic exercises, including jogging, swimming, cycling, walking, gardening, and dancing, have been proved to reduce anxiety and depression.3 These improvements in mood are proposed to be caused by exercise-induced increase in blood circulation to the brain and by an influence on the hypothalamic-pituitary-adrenal (HPA) axis and, thus, on the physiologic reactivity to stress.3 This physiologic influence is probably mediated by the communication of the HPA axis with several regions of the brain, including the limbic system, which controls motivation and mood; the amygdala, which generates fear in response to stress; and the hippocampus, which plays an important part in memory formation as well as in mood and motivation.
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>https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1470658/#i1523-5998-8-2-106-b3

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We have studies showing that exercise is just as effective as medicine in treating depression

>Four trials (n = 300) compared exercise with pharmacological treatment and found no significant difference (SMD -0.11, -0.34, 0.12). From https://pubmed.ncbi.nlm.nih.gov/24026850/

Exercise has massive effects on mitochondria, which might be partly a mechanism in relation to depression. >Mitochondria Linked to Major Depression in Older Adults https://today.uconn.edu/2023/02/mitochondria-linked-to-major-depression-in-older-adults/#>

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Depression is linked to lower brain volume

>Conclusion: Our results suggest that lower CV fitness and exaggerated exercise BP and HR responses in middle-aged adults are associated with smaller brain volume nearly 2 decades later. Promotion of midlife CV fitness may be an important step towards ensuring healthy brain aging.
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>https://n.neurology.org/content/86/14/1313.short?rss=1

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>We found that longitudinal measures of cortical atrophy were widely correlated with sleep quality. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4162301/

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>A better diet quality is associated with larger brain tissue volumes.
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>https://pubmed.ncbi.nlm.nih.gov/29769374/

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>This study shows that a more severe and chronic lifetime disease course in MDD is associated with reduced volume in brain regions relevant for executive and cognitive functions and emotion regulation in a large sample of patients representing the broad heterogeneity of MDD disease course. https://onlinelibrary.wiley.com/doi/10.1002/da.23260>

Sleep is really important, if you aren't sleeping properly you have have a tenfold higher risk of depression,

>People with insomnia , for example, may have a tenfold higher risk of developing depression From https://www.hopkinsmedicine.org/health/wellness-and-prevention/depression-and-sleep-understanding-the-connection

Then there is diet

>The diet may have a significant effect on preventing and treating depression for the individual. A diet that protects and promotes depression should consist of vegetables, fruits, fibre, fish, whole grains, legumes and less added sugar, and processed foods. In the public health nurse’s preventative and health-promoting work, support and assistance with changing people’s dietary habits may be effective in promoting depression. From https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7084175/

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>Current evidence supports the finding that omega-3 PUFAs with EPA ≥ 60% at a dosage of ≤1 g/d would have beneficial effects on depression Https://www.nature.com/articles/s41398-019-0515-5