The affected funds are the European, Japan and US Max Alpha Funds, worth £0.6m, £0.4m and £1.8m respectively, and the £3.8m Green Planet Fund.
A statement released by the firm said the size of the funds meant that they were no longer economical to run and they attracted lower than expected investor demand.
A firm spokesperson added that investors were given the opportunity to transfer into another Neptune fund free of charge, or sell their holdings in March.
European Equities Rob Burnett managed the European Max Alpha Fund and continues to run the European Opportunities fund.
Investment Director Chris Taylor ran the Green Planet and Japan Max Alpha funds and continues to run the Japan Opportunities Fund.