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\section{Introduction}  Our %Our  curiosity about Neanderthals has been increasing steadily during the last 20 years, with 2055 papers already published (12 papers in 1994; 215 papers in 2014; source: Scopus, accessed March 27th, 2015). Our current knowledge is becoming deeper in areas like chronology \cite{Hardy_2010, Gaudzinski_Windheuser_2014, Higham_2014}, ecology \cite{Stewart_2005, Finlayson_2007, Henry_2010}, population dynamics \cite{Trinkaus_1995, Premo_2008, Fabre_2009, S_rensen_2011, Bocquet_Appel_2013}, adaptive traits \cite{Sorensen_2001, S_rensen_2009, Rae_2011}, diet \cite{Stringer_2008, Richards_2009, Henry_2010, Hardy_2011}, technology and cognition \cite{Wynn2004467, Hardy201323, Fern_ndez_Peris_2012, Riel_Salvatore_2010}, genetics \cite{Pr_fer_2013, Briggs_2009, Sanchez_Quinto_2014}, and their relationship with anatomically modern humans \cite{Gibbons_2014, Sankararaman_2012,Banks_2008, d_Errico_1998}. Our knowledge about Neanderthals has greatly increased over the last two decades with more than two thousand research papers published in areas as diverse as chronology \cite{Hardy_2010, Gaudzinski_Windheuser_2014, Higham_2014}, ecology \cite{Stewart_2005, Finlayson_2007, Henry_2010}, population dynamics \cite{Trinkaus_1995, Premo_2008, Fabre_2009, S_rensen_2011, Bocquet_Appel_2013}, adaptive traits \cite{Sorensen_2001, S_rensen_2009, Rae_2011}, diet \cite{Stringer_2008, Richards_2009, Henry_2010, Hardy_2011}, technology and cognition \cite{Wynn2004467, Hardy201323, Fern_ndez_Peris_2012, Riel_Salvatore_2010}, genetics \cite{Pr_fer_2013, Briggs_2009, Sanchez_Quinto_2014}, and their relationship with anatomically modern humans \cite{Gibbons_2014, Sankararaman_2012,Banks_2008, d_Errico_1998}.  Still, our knowledge about their geographical distribution across time and the environmental factors limiting it remains sparse and shallow, and only a few authors have tried to go beyond local or regional presence patterns. For example, \cite{Wenzel_2007} \citet{Wenzel_2007}  describes the distribution of Neanderthals during MIS 5e across central Europe, and discusses briefly the influence of climatic conditions and habitat features over the general occupancy pattern, but does not establish any quantitative link between presence and environment. \citet{Banks_2008} analyzed the distribution of Neanderthals between 43.3 and 36.5 ka BP applying pSDMs, assessed niche conservatism, and analyzed the importance of climatic predictors, concluding that temperature was the most important driver shaping the distribution of this hominin. In this paper we propose a hypothesis about the abiotic drivers of Neanderthals distribution deeply rooted on the Grinnellian niche concept \cite{Grinnell_1917}, the hierarchical framework to explain the importance of species-distribution drivers at different scales proposed by \cite{Pearson_2003}, and our current knowledge on the ecology of Neanderthals (see references above). According to \citet{Pearson_2003}, climate influences species distribution at global to regional scales, while the effect of topography is restricted to scales ranging from regional to local. We hypothesize that the northern and southern edges of Neanderthals range were respectively limited by cold winter temperatures and a combination of high temperatures and low water availability during the summer, whereas a high topographic diversity combined with moderate slopes could have favored occupation at the local scale. Winter temperatures at the northern edge could have exert a negative impact over Neanderthals populations in two related ways: 1) lower availability of small and big game than in temperate areas \cite{Badgley_2000} would have compromised the high caloric intake required by this species \cite{Steegmann_2002, S_rensen_2009}; 2) increased cold stress worsened by a low-caloric diet would have lead to an even more reduced fertility \cite{Bocquet_Appel_2013} and a higher mortality rate \cite{Steegmann_2002}, compromising population survival. Summer temperatures linked to continentality and the higher solar radiation of lower latitudes could have prevented the occupation of southern plains in the Mediterranean peninsulas due to increased heat stress, specially considering the low surface area/volume ratio of this species \cite{Churchill2006}, that hampers heat dissipation. Also, summer drought could have probably reduced access to plant resources, making these areas unsuitable during the warm season. Mediterranean coastal areas could have been suitable because of the buffering effect of the sea over temperature, and the permanent availability of resources like shellfish \cite{Hardy_2011}. At the local scale, high topographic diversity, that fosters biodiversity by an increased availability of ecological niches \cite{Tews_2003}, provided the required abundance and diversity of preys and shelter \cite{Daujeard_2012}, but moderate to low slopes may have been important to reduce the high energetic expenditure of mobility in steep terrain. This limitation, combined with the lower abundance of animals at higher elevations \cite{Brown_2001} could have prevented Neanderthals presence in the higher elevations of the European mountain ranges. 
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In summary, in this paper we propose a hypothesis about how abiotic drivers (climate and topography) could have shaped Neanderthals distribution throughout Europe, and we test it by fitting and analyzing a PSDM describing their distribution during the Last Interglacial.