For a systematic overview of theoretical approaches that link a person's continued existence to psychological connections to the past, see the pertinent description in:

Parfit, D. (1984). Reasons and Persons. Oxford University Press. (See esp. pp. 204–209)

Cf. on approaches in which the continuity of persons is determined by physical or neuronal criteria, e.g:

Fuchs, T. (2018). Leiblichkeit und personale Identität in der Demenz. Deutsche Zeitschrift für Philosophie 66(1), 48–61. https://doi.org/10.1515/dzph-2018-0005 (see esp. pp. 58)

Mc Mahan, J. (2002). The Ethics of Killing. Problems at the Margins of Life. Oxford University Press. https://doi.org/10.1093/0195079981.001.0001(see  esp. pp. 66–69)

Cf. for further reading on the question of how the problem of continuity of dementia patients relates to the question of the binding force of their living wills:

Schmidhuber, M. (2013). Überlegungen zu den Grenzen der Patientenverfügung für die Selbstbestimmung von Demenzbetroffenen im Anschluss die Dworkin-Dresser-Debatte. In J. S. Ach (Eds.), Grenzen der Selbstbestimmung in der Medizin (pp. 317–334). (See esp. pp. 324f.)