Some more from Lazaridis:
"West Eurasians harbour significantly less Neanderthal ancestry than East Asians19-21, whichcould be explained if West Eurasians (but not East Asians) have partial ancestry from asource diluting their Neanderthal inheritance22. Supporting this theory, we observe anegative correlation between Basal Eurasian ancestry and the rate of shared alleles withNeanderthals19 (Supplementary Information, section 5; Fig. 2). By extrapolation, we inferthat the Basal Eurasian population had lower Neanderthal ancestry than non-Basal Eurasianpopulations and possibly none (ninety-five percent confidence interval truncated at zero of0-60%; Fig. 2; Methods). The finding of little if any Neanderthal ancestry in Basal Eurasianscould be explained if the Neanderthal admixture into modern humans 50,000-60,000 yearsago11 largely occurred after the splitting of the Basal Eurasians from other non-Africans.It is striking that the highest estimates of Basal Eurasian ancestry are from the Near East,given the hypothesis that it was there that most admixture between Neanderthals and modernhumans occurred19,23. This could be explained if Basal Eurasians thoroughly admixed intothe Near East before the time of the samples we analyzed but after the Neanderthaladmixture. Alternatively, the ancestors of Basal Eurasians may have always lived in the NearEast, but the lineage of which they were a part did not participate in the Neanderthaladmixture"
"A population without Neanderthal admixture, basal to other Eurasians, may have plausiblylived in Africa. Craniometric analyses have suggested an affinity between the Natufians andpopulations of north or sub-Saharan Africa24,25, a result that finds some support from Ychromosome analysis which shows that the Natufians and successor Levantine Neolithicpopulations carried haplogroup E, of likely ultimate African origin, which has not beendetected in other ancient males from West Eurasia (Supplementary Information, section6) 7,8. However, no affinity of Natufians to sub-Saharan Africans is evident in our genomewideanalysis, as present-day sub-Saharan Africans do not share more alleles with Natufiansthan with other ancient Eurasians (Extended Data Table 1). (We could not test for a link topresent-day North Africans, who owe most of their ancestry to back-migration fromEurasia26,27.) The idea of Natufians as a vector for the movement of Basal Eurasian ancestryinto the Near East is also not supported by our data, as the Basal Eurasian ancestry in theNatufians (44±8%) is consistent with stemming from the same population as that in theNeolithic and Mesolithic populations of Iran, and is not greater than in those populations(Supplementary Information, section 4). Further insight into the origins and legacy of theNatufians could come from comparison to Natufians from additional sites, and to ancientDNA from north Africa. "
This is why I started to consider perhaps a movement from India into Mesopotamia and then a spread from there.