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, esp. 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, esp. 58. https://doi.org/10.1515/dzph-2018-0005

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

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, esp. 324 f.).